Class 
Book 




>-\ * 



Copyright N° 

COPYRIGHT DEPOSIE 



The Place of the Elementary Calculus in 
the Senior High-School Mathematics 

And Suggestions for a Modern 
Presentation of the Subject 



By 

NOAH BRYAN ROSENBERGER, Ph.D. 
U 



Teachers College, Columbia University 
Contributions to Education, No. 117 



Published by 

tEeadjera College, Columbia SUmbersttp 

New York City 
1 9 2 1 



CO 



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Copyright, /921, by Noah Bryan Rosenberger 



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OCT 14 1922 
C1A686289 



PREFACE 

Even a little less than a decade ago, the trained teacher of high- 
school mathematics had the pleasure of teaching real mathematics 
as found in such subjects as trigonometry, solid and plane geometry, 
and advanced and elementary algebra. It is doubtless true that this 
kind of work was inclining too much toward formalism and that the 
needs of the learner were not being given sufficient attention. The 
strong reaction against formalism in general in education during 
the decade just past made the position of any genuine mathematics 
in our high schools very precarious. This threatened elimination of 
all real mathematics in our high schools was viewed with regret by 
those of us who had had the pleasure of starting many a pupil with a 
vigorous mind upon his mathematical way; and a number of prom- 
inent educators who are especially interested in the teaching of 
mathematics have been working toward a readjustment of the 
mathematics curriculum so that it will fit into and fulfill its mission 
in the changed conditions as found in our schools to-day. 

The following study is the author's contribution toward this 
attempted readjustment. The kind of mathematics that should be 
taught in the first six school years has been definitely settled; the 
main points of the work in mathematics that should be included in 
the junior high-school period have been agreed upon; and the 
author hopes that his study will be of some value in helping us to 
formulate the content of the mathematics curriculum of the senior 
high school. 

The author takes pleasure in acknowledging his indebtedness to 
Professor David Eugene Smith for his wise and constructive criti- 
cism and helpful guidance in the preparation of this study. 

Noah Bryan Rosenberger 

Acting Head of Department of Mathematics, 
Dickinson College, Carlisle, Pa. 
July i, 1921 



TABLE OF CONTENTS 

Page 

Introduction i 

I. A Study of the Status of Mathematics in the Schools Abroad 

that Correspond to our High Schools 5 

As indicated by their textbooks and schools, English, French, and 

German 5 

As indicated by the courses in analytic geometry and in the calculus 

in different countries 7 

The year corresponding to our tenth school year 7 

The year corresponding to our eleventh school year 7 

The year corresponding to our twelfth school year 8 

Conclusions 8 

II. The Trend of Mathematics in our Public School System ... 10 

A reason for the reorganization in the elementary-high-school period 10 

Junior high-school mathematics * 10 

Senior high-school mathematics 11 

Mathematics elective 11 

Various courses 12 

The tenth school year 12 

The eleventh school year 13 

The twelfth school year 14 

A sufficient number of teachers available 15 

Conclusions 16 

III. The Important Position Occupied by the Calculus in the Math- 
ematics Structure 18 

The wide application of the topic of maxima and minima indicated 19 

Conclusions 19 

IV. A Historical Survey of the Natural Growth of the Calculus 

in the Development of Mathematics 20 

Reasons for making this historical survey 20 

The period preceding Isaac Barrow 21 

The first general step in the development of the calculus .... 21 

A bit of integration 21 

The second general step in the development of the calculus ... 22 

Barrow, Newton, and Leibniz 23 

Isaac Barrow 23 

Isaac Newton 24 

Gottfried Wilhelm Leibniz 25 

The early textbooks on the calculus 25 

Jean Bernoulli 25 

Marquis de l'Hospital . . 28 



vi Contents 

Charles Hayes 28 

Christian Wolf 30 

Leonhard Euler 31 

M. Cousin 31 

Lazare Carnot 31 

Augustin Louis Cauchy 31 

Conclusions 32 

V. Comparison of Textbooks on the Elementary Calculus for 

Beginners and for Self-instruction 33 

The standard method 33 

Textbooks reviewed 34 

American 34 

English 34 

French 36 

German 37 

Conclusions 37 

VI. The Trend of American Education in General 39 

The increased length of school attendance 39 

The changed point of view in education 40 

The views of others 41 

The need of more trained mathematical minds and the source of 

supply 41 

Conclusions 43 

VII. Suggestions for a Modern Presentation of the Elementary 

Calculus 44 

The aim of these suggestions 44 

The divisions of the discussion 44 

First Part 45 

General features influencing the plan of the course 45 

The viewpoint 45 

Use 45 

Appreciation 46 

Disciplinary value 46 

Interest 47 

Degree of difficulty 47 

The aim of the course in the elementary calculus in the senior 

high-school mathematics 48 

The method of presentation 48 

The material 49 

The use of graphs 49 

Second Part 50 

The method of approach to the subject 50 

Probable methods 50 

The method recommended 50 

What is the calculus? 51 



Contents vii 

A reason for studying the calculus 51 

The previous work in graphs 51 

The function idea 53 

Previous training of the pupil in the function idea 53 

Illustration of the function idea 53 

Independent and dependent variables. Definition of a function 55 

Increment 55 

Function notation 56 

Numerical illustration 56 

Increment illustration 57 

Other ways of representing the function besides f(x) 58 

The derivative 59 

The first step 59 

The symbol /' (x) 60 

The second step 61 

The fundamental rule 62 

Application of the fundamental rule 62 

Remarks 63 

The slope of the tangent 63 

Formulas 65 

The differentiation of the function ku with respect to x . . . 65 

Applications of the derivative 65 

Maxima and minima 66 

A minimum in a graph 66 

A minimum illustrated by a practical problem 67 

A maximum in a graph 68 

A maximum illustrated by a practical problem 69 

Maxima and minima in graphs 70 

Suggestions for plotting a curve 71 

The differential 72 

Reasons for studying the differential 72 

Geometrical representation 72 

What dy equals 72 

Integration 73 

A reason for studying integration 73 

Integration the reverse of differentiation 74 

A first attempt at a table of integrals 75 

The definite integral 76 

The length of an arc 77 

The formula 77 

An application of the formula 78 

The volumes of certain solids 79 

The formula 79 

An application of the formula 80 

General Conclusions 81 



INTRODUCTION 

A study of the nature of the courses in mathematics found from 
time to time in our high schools will show marked changes in the 
kind and extent of the topics included in these courses. These 
changes will be indicated in a general manner in the discussion 
which follows. In colonial times, in the schools corresponding 
approximately to our high schools, the arithmetic consisted of 
operations with integral numbers and of some work in fractions and 
in proportion. A few decades later, when our high schools first began 
to make their appearance, arithmetic had become more difficult 
and included such topics as stocks and exchange, circulating deci- 
mals, alligation, cube root, and permutations and combinations. 
Then it was realized that it would be better not to teach so much of 
the difficult parts of arithmetic but more of the simpler parts of 
some of the higher branches in mathematics. Hence, during the 
latter part of the nineteenth century, algebra and geometry became 
a part of the high-school mathematics. 1 These were followed a 
number of years later by the introduction of trigonometry. 

In our present decade there is taking place an extended reorgani- 
zation in our school curricula. One of the phases of this reorganiza- 
tion, namely, the junior high-school movement, has caused a radical 
change in the content of the mathematics curriculum. It has 
brought into the courses in mathematics in the seventh, eighth, and 
ninth school years the simpler parts of geometry, algebra, and 
trigonometry. This makes necessary an entire rearrangement of the 
courses in mathematics in the tenth, eleventh, and twelfth school 
years, or the senior high-school period. Such subjects as demonstra- 
tive geometry, algebra, and trigonometry are assured of a place in 
the mathematics of the senior high school. However, these sub- 
jects will not now be sufficient to meet the needs of pupils who wish 
to elect courses in mathematics in each of these three years. These 
needs will probably be met, as similar ones have been in the past, by 
the introduction of the simpler parts of some of the higher branches 

1 Paul Monroe (ed.), A Cyclopedia of Education, Vol. 3, pp. 263 and 267, New York, 
1912; Florian Cajori, The Teaching and History of Mathematics in the United States, 
p. 9, Washington, 1890. 



2 Elementary Calculus in Senior High-School Mathematics 

in mathematics ; and it is the aim of this study to show that among 
these should be included the simpler parts of the calculus. 



GENERAL STEPS IN THE ARGUMENT 

I. The first topic will be a discussion based upon a study of the 
status of mathematics in foreign schools corresponding to our high 
schools. This will show that in many countries the calculus is taught 
in these schools before the end of the year which corresponds to our 
twelfth school year. This study will also give us an outside standard 
whereby to judge the status of mathematics in our schools. 

II. The second topic will be a discussion of the trend of mathe- 
matics in our public school system. This will show that there has 
been created an opportunity for including the elementary calculus 
in the senior high-school mathematics. The question then arises, 
since there is an opportunity for doing this, is the subject of suffi- 
cient importance? 

III. The third topic will be a discussion of the important position 
occupied by the calculus in the mathematics structure. This will 
show that the calculus is one of the most important aids in applied 
mathematics and contributes largely to the field of pure mathe- 
matics. The conclusion will be drawn that, because of its impor- 
tance, senior high-school pupils who are interested in mathematics 
should be given an opportunity to become acquainted with the 
elementary calculus, provided it is not too difficult. 

IV. The fourth topic will give a historical survey of the natural 
growth of the calculus in the development of mathematics. This 
will show that a first course in the elementary calculus is well within 
the mental grasp of pupils in mathematics in the senior high school. 
This historical survey will also suggest suitable methods of teaching 
the elementary calculus to younger pupils. 

V. The fifth topic will give some of the features of various text- 
books on the elementary calculus that have been written for begin- 
ners. It will show that a number of professors and able teachers of 
mathematics do not consider the elementary calculus too difficult for 
younger pupils. This comparison of textbooks will also be of value 
in the selection of methods of presentation of the elementary cal- 
culus. 

VI. The sixth topic will indicate the trend of American education 



Introduction 3 

in general. It will show that among many people educational ac- 
tivity does not cease at the close of their public school life. Of course 
these later mental pursuits follow many different channels. A per- 
son working in the electrical engineering field will be likely to take 
up studies along the lines of electricity and engineering. There are 
many such fields in which a further study of the calculus will be an 
advantage. The course in the elementary calculus in the senior 
high school will make it possible for the pupil who has taken it to 
continue further his study of the calculus without the aid of a 
teacher. 

VII. The seventh topic will give suggestions for a modern presen- 
tation of the elementary calculus. These suggestions will show how 
to simplify the subject, make the topics definite, and lead the 
pupils to see the theory as a related whole. 

Among the sources from which some of these suggestions have 
been derived is the historical survey of the calculus in the develop- 
ment of mathematics. A comparison of textbooks on the ele- 
mentary calculus which have been written for beginners has also 
furnished a means for finding out some of the most promising 
methods. 

An additional source for these suggestions is the author's own 
experience. For several years he gave a course in advanced algebra 
to the scientific group in the public high school of a third-class city 
in which was included a considerable amount of work in differentia- 
tion. This work in differentiation included the greater part of the 
course in the differential calculus for use in the senior high school as 
it is outlined in the discussion under this section. His experience in 
teaching the calculus to classes in college has enabled him to deter- 
mine the explanations necessary to bridge over the gap between the 
student's understanding and the usual method of presenting the 
calculus in college. The general training that he received in teach- 
ing pupils of a different race and language in the Orient has also 
been an aid to him in making troublesome topics clear to American 
pupils. 

The author's graduate studies in pure mathematics have fur- 
nished him with a criterion for developing the course in the ele- 
mentary calculus from the standpoint of mathematical rigor; that 
is, the deductions made throughout the course are sound, although 
their proofs may not be given. On the practical side, he has had the 



4 Elementary Calculus in Senior High-School Mathematics 

benefit derived from the study of the mathematics included in a 
college course in electrical engineering. This benefit has been in- 
creased by his association with managers of a large steel plant and 
of a large electrical firm. 

The author is also indebted for suggestions to co-workers at Teach- 
ers College to whom he had submitted his manuscript. 2 He also 
profited by the Round Table Conference in Mathematics where he 
presented a part of this manuscript. 3 

2 Among these are Messrs. M. A. Norgaard, assistant professor of mathematics, 
Grinnell College; J. R. Clark, editor of The Mathematics Teacher; W. W. Rankin, 
professor of mathematics, Agnes Scott College, Georgia; R. Beatley, of the department 
of mathematics of the Horace Mann School for Boys; and Misses H. Calkins, of the 
faculty of mathematics of Knox College; P. Wiggins, of the Los Angeles City Schools; 
T. Schlierholz, of the Ben Blewett Junior High School, St. Louis; R. S. Nichols, of the 
Vail-Deane School; M. A. Weber, of the Saginaw East Side High School, Michigan. 

3 Round Table Conference, Department of Mathematics, Teachers College, Colum- 
bia University, March 19, 1921. 



I 

A STUDY OF THE STATUS OF MATHEMATICS IN THE 

SCHOOLS ABROAD THAT CORRESPOND TO OUR 

HIGH SCHOOLS 

AS INDICATED BY THEIR TEXTBOOKS AND SCHOOLS 

The status of mathematics in the schools abroad that correspond 
to our high schools will be indicated principally by a discussion on 
the extent of work covered by these schools in the subjects of ana- 
lytic geometry and the calculus. A comparison of the leading Euro- 
pean textbooks on the elementary calculus shows that this subject 
is taught to younger pupils than is customary in our American 
schools. (A more extended discussion on a comparison of these 
textbooks is given in Section V.) The German textbooks on the 
elementary calculus which have been written for use in the Prussian 
secondary schools adhere rather closely to the scientific method. 
In France, the elementary calculus forms a part of their course in 
algebra. The French stress generalization rather than application. 
However, the English authors of this type of book have made a 
radical attempt to treat the subject from the point of view of the 
young student. The author of one of these recent English text- 
books, who is himself a teacher of mathematics for engineering 
students, says in the preface, "Fortunately, the calculus is now be- 
ginning to be taught to boys of fifteen or sixteen in day and even- 
ing technical schools, and it will grow up with them in the same 
way as their trigonometry and arithmetic have done." l 

In the Manchester Grammar School (England) the curriculum 
for the Mathematical Sixth includes algebra, trigonometry, analytic 
geometry, and differential and integral calculus. This work corre- 
sponds to the work in mathematics given in the freshmen and 
sophomore years in American colleges. 2 

In the Lycees (France), the course in secondary instruction covers 
a period of seven years and is coordinate with the preceding pri- 

1 John Stoney, An Introduction to the Differential and Integral Calculus. (Junior 
Day Technical School and Municipal Technical School, Smethwick, London.) 

2 Classical and Science Lists of The Manchester Grammar School, for the Midsummer, 
IQ20. Furnished by Dr. I. L. Kandel, of Teachers College, Columbia University. 



6 Elementary Calculus in Senior High-School Mathematics 

mary instruction which covers a period of four years. 3 The period of 
secondary instruction is divided into two cycles ; the first cycle con- 
sists of four years and the second one of three years. In the first year 
(which corresponds to our eleventh school year) of this second 
cycle, the course in algebra for the Latin- Sciences and the Sciences- 
Langues vivantes sections includes the following work in the calculus : 
the notion of a derivative, geometric signification of the derivative, 
the sense of the variation indicated by the sign of the derivative, 
applications to very simple numerical examples and in particular to 
functions in preceding studies. In the next year in these two sec- 
tions, the calculus is extended and includes the calculation of 
derivatives of the simple functions and a study of rectilinear motion 
by means of the theory of derivatives. In the last year, the work in 
the classes in mathematics includes the derivative of a sum, product, 
quotient, of the square root of a function, of sin x, of cos x, of tan x, 
and of cotg x, the topics of maxima and minima, and of the area 
of a curve regarded as a function of the abscissa. The calculus in the 
special class in mathematics is equivalent to that of our usual col- 
lege course in the subject. 

The reform movement in mathematics in the Oberstufe of Ger- 
many had the function concept as its central idea. 4 The following 
divisions into groups shows how the schools reacted toward this 
idea. 

a. Those schools which declined to adopt the idea of the func- 
tion concept. 

b. Those schools which believed in a late discrete development 
of the function idea: without the infinitesimal calculus; with the 
differential calculus; with the differential and integral calculus. 

c. Those schools which believed in a gradual, continuous develop- 
ment of the function concept: without the infinitesimal calculus; 
with the differential calculus in the Prima (the last year) ; with the 
differential and integral calculus already in the Obersekunda (the 
second half of the year preceding the last year). 

The nature of the work in the calculus in the hbheren Schulen in 
Germany is given under the next topic. 

3 Delalain Freres (ed.). Plan D' Etudes et Programmes D'Enseignement dans les Lycees 
et Colleges de Garcons, p. xviii, Paris, 1907-1908. F. E. Farrington, French Secondary 
Schools, New York, 191 o. 

4 Walter Lietzmann, Die Organisation des Mathematischen Unterrichts an den Ho- 
heren Knabenschulen in Preussen, p. 199, Leipzig, 1910. 



Status of Mathematics in Schools Abroad 7 

AS INDICATED BY THE COURSES IN ANALYTIC GEOMETRY AND THE 
CALCULUS IN DIFFERENT COUNTRIES 

This discussion is based upon the reports of the International 
Commission on the Teaching of Mathematics. This Commission 
owes its origin to the International Congress of Mathematicians at 
Rome in 1908. Thus far eighteen countries have made reports. 
These are so extensive that they cover over 12,000 pages. 5 

The work in these two courses, analytic geometry and the cal- 
culus, in the foreign schools for the years that correspond to our 
tenth, eleventh, and twelfth school years will now be considered. 
The material for this comparison has been selected from the United 
States Bureau of Education Bulletin No. 45, which gives in syste- 
matic form that part of the report of the International Commission 
on the Teaching of Mathematics that relates to curricula in mathe- 
matics. 6 

THE YEAR CORRESPONDING TO OUR TENTH SCHOOL YEAR 

The ages of the pupils in this school year (10) are approximately 
fifteen to sixteen years. In this year in the Realschule of Russia, 
both plane analytics and the elements of the infinitesimal calculus 
are included in the course in mathematics. In analytic geometry, 
the principal theorems are derived by means of rectangular coordi- 
nates, and the equations for the circle, ellipse, parabola, and hyper- 
bola are derived both in rectangular and polar coordinates. Their 
infinitesimal calculus is of the nature of an introductory course in 
the calculus. The simpler formulas for the derivative are taught. 
The work also includes the topics of maxima and minima, and in- 
definite and definite integrals. 

In this year (10) the analytic geometry is also taught in the 
Realgymnasium and Oberrealschule of Germany. 

THE YEAR CORRESPONDING TO OUR ELEVENTH SCHOOL YEAR 

The ages of the pupils in this school year (11) are approximately 
sixteen to seventeen years. An introductory course in analytic 

5 Raymond Clare Archibald, The Training of Teachers of Mathematics, United 
States Bureau of Education Bulletin, No. 27, 191 7. 

8 J. C. Brown, Curricula in Mathematics, United States Bureau of Education Bulle- 
tin, No. 45, 1914. 



8 Elementary Calculus in Senior High-School Mathematics 

geometry is given in this year (n) in the Gymnasium in Austria, 
and a more difficult course in the Realschule. In the latter school, 
easy differential and integral calculus are introduced in order to 
simplify or make more intensive the knowledge of physics. 

In this year (n) in the Oberrealschule in Germany, courses in 
analytic geometry and in the differential calculus are included in 
the work in mathematics, but the calculus is not taught in this 
year in the Gymnasium and the Realgymnasium. A course in ana- 
lytic geometry is also included in this year in the curricula of the 
three main schools of Austria, of the Lycees of France, and of the 
Realschule and Gymnasium of Hungary, and of the Realgymnasium 
of Sweden. 

THE YEAR CORRESPONDING TO OUR TWELFTH SCHOOL YEAR 

The ages of the pupils in this school year (12) are approximately 
seventeen to eighteen years. In this year, analytic geometry is 
taught in the Gymnasium schools of Holland, and of Hungary. 
Also, in this school year which corresponds to our twelfth school 
year, both analytic geometry and the calculus are taught in Austria, 
Belgium, Denmark, England, France, Germany, Roumania, 
Sweden, and Switzerland. The work in these two subjects ranges 
from that of a simple introduction to a course equivalent to our first 
course in college. 

CONCLUSIONS 

Thus, it is evident that the United States is far behind other 
countries in the work that our high schools do in mathematics. 7 
That this difference has not appeared more prominently heretofore 
in such fields as engineering is partly due to the fact that our larger 
quantity of superior native ability has been able to compensate this 
difference. Owing to the rapid expansion of all fields making use of 
mathematics, however, and to the stronger competition resulting 
from the closer relations of the nations of the world both as to the 
shortening of time in transportation and as to political interests, 
the need of more men with trained mathematical minds cannot be 
met by relying upon native ability alone but by giving this native 

7 Even the conditions in our own colleges show that the calculus should be placed 
earlier in the curriculum since frequently engineering students are asked to apply its 
principles in physics and mechanics before these principles have been developed in tlr- 
regular calculus course. 



Status of Mathematics in Schools Abroad 9 

ability mathematical training of the most careful sort. To give this 
training in a satisfactory manner, it must be begun in the senior 
high-school period. 8 

The main reason that pupils in the schools abroad are able to 
accomplish this work in mathematics which throws them about two 
years ahead of our American pupils by the end of the twelfth school 
year is because of the more profitable arrangement of the courses in 
the mathematics curriculum in their schools. A more profitable 
arrangement has already been made in the mathematics curriculum 
of the American standard junior high school, and an attempt is now 
being made to do a similar thing for the mathematics curriculum 
of our senior high school. In the junior high school, the propaedeutic 
courses in algebra, geometry, trigonometry, and graphs lay the 
foundations for the senior high-school courses in additional algebra, 
demonstrative geometry, additional trigonometry, and the intro- 
ductory work in analytic geometry. Thus, the foundation is laid 
for an introductory course in the elementary calculus in the twelfth 
school year. A scheme such as this would furnish an opportunity to 
our pupils who have the inclination and the native ability to com- 
pete favorably with the highly trained pupils abroad, and would aid 
us as a nation in maintaining a foremost place in the fields of applied 
science and of engineering. 

8 "If the student who omits the mathematical courses has need of them later, it is 
almost invariably more difficult and it is frequently impossible for him to obtain the 
training in which he is deficient. In the case of a considerable number of alternative 
subjects a proper amount of reading in spare hours at a more mature age will ordinarily 
furnish him the approximate equivalent to what he would have obtained in the way of 
information in a high-school course in the same subject. It is not, however, possible 
to make up deficiencies in mathematical training in so simple a fashion. It requires 
systematic work under a competent teacher to master properly the technique of the 
« subject and any break in the continuity of the work is a handicap for which increased 
maturity rarely compensates. Moreover, when the individual discovers his need for 
further mathematical training, it is usually difficult for him to find the time from his 
other activities for systematic work in elementary mathematics." — Elective Courses 
in Mathematics for Secondary Schools, a preliminary report by The National Com- 
mittee on Mathematical Requirements. 



II 

THE TREND OF MATHEMATICS IN OUR PUBLIC SCHOOL 

SYSTEM 

A REASON FOR THE REORGANIZATION IN THE ELEMENTARY-HIGH- 
SCHOOL PERIOD 

The mathematics curriculum of the first nine school years has 
already adapted itself to the changed conditions due to the reor- 
ganization in the elementary-high-school period. "One of the main 
reasons for taking two years from the elementary-school period and 
adding them to the high-school period, thus dividing the elementary- 
high-school period of twelve years into a period of six years of ele- 
mentary school and six years of high school, and then subdividing 
the high-school period into three years of junior high school and 
three years of senior high school, was to make possible a better 
adjustment of studies and school methods in the three years now 
coming to be quite commonly included in the junior high school." 1 

JUNIOR HIGH-SCHOOL MATHEMATICS 

The nature of the mathematics in the three years of the junior 
high school can probably be most simply illustrated by an outline 
of the contents of a suitable set of textbooks on junior high-school 
mathematics. There are a number of such sets of textbooks on the 
market. The following is the outline of the contents of one of these : 

First Year 

First half. Arithmetic. 

Arithmetic of the home, the store, the farm; arithmetic of industry; arith- 
metic of the bank; material for daily drill. 

Second half. Geometry. 

Geometry of form; geometry of size; geometry of position; supplementary 
work; tables for reference. 

Second Year 

First half. Algebra. 

The formula; the equation; the graph; negative numbers; algebraic oper- 
ations; further uses of algebra. 

1 United States Bureau of Education, Secondary School Circular, No. 6, July, 1920. 
Statement by Commissioner P. P. Claxton. 



Trend of Mathematics 1 1 

Second half. Arithmetic. 

Arithmetic of trade, of transportation, of industry, of building; arithmetic 
of the bank, of corporations; arithmetic of home life, of the farm, of commu- 
nity life, of civic life; arithmetic of investments, of mensuration; material 
for daily drill; tables for reference. 

Third Year 

First part. Algebra. 

The chief uses of algebra; addition and subtraction; multiplication; divi- 
sion; fractions; simple equations; quadratic equations; general review. 

Second part. Trigonometry. 

Functions of angles, with applications; trigonometric tables, with applica- 
tions. 

Third part. Demonstrative geometry. 

Meaning of geometry; triangles; parallel lines; parallelograms; angle 
sums; areas; constructions; loci; review; table for reference. 2 

According to the arrangement of the work in mathematics in the 
standard junior high school, propaedeutic courses have been given 
in algebra, geometry, and trigonometry by the end of the ninth 
school year. The nature of the subject-matter in these three 
branches differs from the customary textbook type. Instead of an 
unnecessary amount of time and space being given to puzzle prob- 
lems, problems of mere complexity, and problems of a too advanced 
character merely to make the work exhaustive, the time and space 
are devoted to a simple, careful, and helpful introduction into 
these subjects. 

Now that the mathematics curriculum of the junior high school 
has been made definite, it can be used as a basis upon which to build 
the mathematics curriculum of the senior high school. 

SENIOR HIGH-SCHOOL MATHEMATICS 3 

Mathematics Elective. It cannot be expected nor would it be advis- 
able that mathematics should be required of all pupils in the senior 
high school. One of the reasons for this is that there are some pupils 
who have not minds capable of the amount of rigorous thinking 
required in mathematics. These pupils will not elect mathematics, 
and those who do elect mathematics will do so largely because they 
like the subject. For this reason, there will not be the need of arti- 

2 Wentworth-Smith, Junior High-School Mathematics, New York. 

3 Some of these ideas were expressed by Professor D. E. Smith in his lectures at 
Teachers College, February, 192 1. 



12 Elementary Calculus in Senior High-School Mathematics 

ficial motivation. Hence, we can follow the shorter road into 
mathematical insight by making use of suitable material that 
may not have any practical application whatever, and the pupils 
will enjoy it. There will not then be the necessity for adapting 
mathematics to mathematically weak minds since these pupils 
will not take mathematics. This selection will give us the stronger 
type of mathematical mind and that strong kind of incentive which 
comes from pupils who have mental hunger for this particular line 
of study. These pupils will take up the subject for the love of it. 
Accordingly, so far as mathematics is concerned, our high schools 
will be better off than corresponding schools abroad, since there 
the selection of pupils is usually on the basis of wealth and not on 
the basis of ability. 

Various Courses. Various courses in which this elective mathe- 
matics may run are vocational, commercial, industrial, and agri- 
cultural courses. Commercial mathematics is such an extensive 
subject that it is taken care of by itself in the commercial high 
schools. There this work covers four years. There is also a great 
deal of mathematics required in industrial work, as is shown by the 
numerous technical schools which exist. There is little mathe- 
matics needed in agriculture. Also, there will be some pupils who 
will want mathematics for informational purposes and some for 
college entrance. 

The possible courses themselves will now be taken up according 
to the school years in the senior high school. 

The Tenth School Year. In the tenth school year, what should 
the student have if he likes mathematics, or if he wants to go to 
college, or if he wants to enter industrial fields where he will find it 
advantageous to continue further his studies in mathematics of 
his own account? There is no question but that he will need to 
take additional work in algebra. This too would be the time for 
him to become further acquainted with those geometric ideas which 
later will make it possible for him to grasp more readily the mathe- 
matical conditions involving such ideas. At the same time, the 
course in plane geometry should be so arranged that it will serve 
one of its main purposes, namely, to furnish the best developed 
applications of logic that we have. Along with this work in plane 
geometry, there should be given at opportune times the related 
notions of solid geometry in order that the learner's mind may not 



Trend of Mathematics 13 

be unnecessarily restricted to a geometric world of two dimensions. 
Such an arrangement will also give the preliminary exercise in the 
development of three dimensional space perception. In fact, there 
is no longer a necessity for teaching solid geometry as a separate 
subject in our high schools, and there is a tendency in a number of 
our high schools toward the elimination of solid geometry as a sepa- 
rate subject. 4 Its elimination can be effected without the loss of its 
three main aims, the development of space perception, the exten- 
sive work in mensuration, and its introductory and almost intuitive 
treatment of incommensurables and limits. It has already been 
suggested that the first of these can be partly taken care of in the 
course in plane geometry. The more extended practice in the de- 
velopment of space perception, the extensive work in mensuration, 
and the treatment of incommensurables and limits can be presented 
in a much more satisfactory manner in a course in the elementary 
calculus adapted for use in the senior high school. 

Hence, the two main courses in mathematics in the tenth school 
year should be algebra and demonstrative geometry. With the 
propaedeutic courses in mathematics in the junior high school as 
a foundation, the minimum time to be devoted to either of these 
two subjects need not be greater than half a school year. 

The Eleventh School Year. The first half of this year could be 
devoted to elementary courses in statistics and descriptive geometry. 
This course in elementary statistics would be a valuable aid to the 
pupil in his general reading. It would enable him to comprehend 
more readily the concentrated information contained in statistical 
charts that are found so frequently in current articles. The course 
in descriptive geometry would aid the pupil in extending his knowl- 
edge concerning the properties of solids and develop skill in drawing 
in perspective. This kind of drawing is the one most used in future 
work. 

The second half of the year could be devoted to work in trigonom- 
etry in addition to the numerical trigonometry given in the junior 
high-school mathematics. In the second half of this year, there 
should also be given a considerable amount of graph work. The 
finding of maxima and minima of quantities by means of graphs 
should be especially emphasized. In rectangular coordinates, the 

4 Questionnaire to Determine the Place of Solid Geometry in the High-School Curricu- 
lum, sent out by the author, spring, 1920. 



14 Elementary Calculus in Senior High-School Mathematics 

equations and some of the simple properties of the straight line, 
the circle, the parabola, the ellipse, and the hyperbola should be 
taught. 

The Twelfth School Year. It is seen that there is room in the 
last year of the senior high school for an additional course in mathe- 
matics. The mathematics of the tenth and eleventh school years 
were preparing the pupil principally to enter one of two fields, 
either the field of pure mathematics or some special field of industry 
in which he would find it desirable to continue further some of his 
studies in mathematics. Is there one such course that will give the 
pupil desiring to enter the industrial field such additional intro- 
duction into the field of mathematics that he will be able thereafter 
to continue his further study in mathematics without the aid of a 
teacher? Some of the possibilities of such a course will now be dis- 
cussed. 

In the twelfth school year, what may be done for the pupil from 
the standpoint of his love for mathematics and of the use it will be 
to him in his general reading? He might be given a more extended 
course in analytic geometry. This latter course does not lead any- 
where in this school year except to analytic geometry since there 
are no applications. He might be given higher algebra, including 
such topics as permutations and combinations, theory of equations, 
and the theory of determinants. It is less than one hundred years 
that people have been able to multiply one determinant by another. 
However, higher algebra also does not lead anywhere. There 
might be included a course in projective geometry. This would 
lead the pupil up upon a high mountain so far as the field of geom- 
etry is concerned, and this would be very interesting. From the 
geometric side alone this is one of the best things that could be 
done but it too has no applications, and hence for a good many 
minds it would not be as interesting as something that has some 
applications. Yet there is a subject that is interesting, that is 
not too difficult, that leads to work having applications, that has 
an important propaedeutic value, and that will deepen the pupil's 
appreciation of mathematics. This subject is an introductory 
course in the elementary calculus. Accordingly, in the last year of 
the senior high school, or the twelfth school year, there should 
be offered as an elective in mathematics an introductory course 
in the elementary calculus. This raises the question, are there a 



Trend of Mathematics 15 

sufficient number of teachers available who are capable of teaching 
this course? 

A Sufficient Number of Teachers Available. The question of ob- 
taining a sufficient number of teachers qualified to teach the ele- 
mentary calculus in the senior high school does not present much 
difficulty. There are a number of teachers now in the service who 
can do this. The later standards of qualifications for teaching will 
increase this number. An illustration of one of these standards is 
as follows : "The minimum attainment of teachers of any academic 
subject shall be equivalent to graduation from a college belonging 
to the North Central Association of Colleges and Secondary Schools 
requiring the completion of a four year course of study or 120 
semester hours in advance of a standard four year high-school 
course." 5 

Such standards as the one just quoted will cause the prospective 
teacher of mathematics to include sufficient mathematics in his 
preparatory work to qualify him to teach the elementary calculus 
of the senior high school. Also, there is the opportunity of teachers 
now in service to prepare themselves for this work by taking work 
in mathematics in summer sessions. "Summer schools are conducted 
by universities and colleges and normal schools, or are specially 
organized by State educational authorities. In general, their work 
is intended for teachers in service. Courses in mathematics have 
been regarded in most cases as of special importance in these sum- 
mer curricula. In fact, in several instances the work began with 
courses in mathematics and perhaps one other subject, and was 
extended to include courses in all the regular departments as the 
number of students increased and the demand became apparent." 6 

The following university standard is additional evidence that 
soon there will be no lack of teachers of mathematics qualified to 
teach the elementary calculus in the senior high school: "The de- 
partment of mathematics at Brown University lays down a mini- 
mum course of study which it requires students who are prospective 

5 Report of North Central Association of Colleges and Secondary Schools, edited by 
C. O. Davis, p. 37-A, 1920. This association also reports seventy-seven bona fide senior 
high schools. This report was obtained through the aid of Provost C. B. Upton, of 
Teachers College. 

6 The Training of Elementary- School Teachers in Mathematics in the Countries repre- 
sented in the International Commission on the Teaching of Mathematics, I. L. Kandel, 
United States Bureau of Education Bulletin, No. 39, p. 53, 1915. 



1 6 Elementary Calculus in Senior High-School Mathematics 

teachers of mathematics to take if they wish the backing of the 
department in starting on their careers. In outline the course is as 
follows: Plane trigonometry (3 semester hours), higher algebra 
(3), solid geometry (3), plane analytic geometry (4), differential 
and integral calculus (8), teachers' course in algebra (6), and 
teachers' course in geometry (6)." 7 

Even the prospective junior high-school teacher of mathematics 
should realize that he will need considerable preparation in pure 
mathematics if he wishes to obtain a proper appreciative attitude 
toward the elementary content of the junior high-school curriculum 
in mathematics. 8 The prospective junior high-school teacher needs 
to have strengthened or even first imparted the following con- 
cepts, skills and information: formula work, graph work, concept 
of the function, trigonometry, computation, drawing, and proper- 
ties of solids. For his basic foundation for formula work, he should 
go to trigonometry; for graph work, to analytic geometry; for the 
concept of the function, to the calculus; for computation, to loga- 
rithms; for drawing and the properties of solids, to descriptive 
geometry. 

That the number of teachers who are college graduates is on the 
increase is indicated by the following quotation from statistics for 
the State of New Jersey for the year ending in June, 1920. "In 
high schools the number of new teachers who were graduates of 
colleges, universities or technical institutions was 89 more than 
last year." 9 

CONCLUSIONS 

The discussion in this topic has shown that the courses in math- 
ematics preceding the twelfth school year not only form a suitable 
preparation for but also leave room for the introduction of the 
elementary calculus in the twelfth school year. 

This discussion also shows that there will be available a sufficient 
number of teachers qualified to teach the elementary calculus in 
the senior high school, since there are a number of such qualified 

7 Bureau of Education Bulletin, No. 27, by Raymond Clare Archibald, Associate 
Professor of Mathematics in Brown University, p. 206, 191 7. 

8 Percival M. Symonds, "Subject Matter Courses in Mathematics for the Pro- 
fessional Preparation of Junior High-School Teachers," Educational Administration 
and Supervision, February, 192 1. 

9 Furnished by Professor T. H. Briggs, Teachers College, Columbia University. 



Trend of Mathematics 1 7 

teachers now in the service, since others will take the opportunity 
to prepare themselves for teaching the elementary calculus, and 
since the future teachers of mathematics will have sufficient math- 
ematics included in their preparatory work to qualify them for 
this purpose. 



Ill 

THE IMPORTANT POSITION OCCUPIED BY THE 

CALCULUS IN THE MATHEMATICS 

STRUCTURE 

Practically all the mathematics that everybody should know is 
included in the curriculum of the first nine school years. There is 
also some elective mathematics offered in the ninth year, or last 
year of the junior high school. It was shown in the discussion 
under the preceding topic that upon this foundation there should 
be built the courses in demonstrative geometry, algebra, and 
trigonometry; and, also, that these courses together with the work 
in graphs form a suitable preparation for the simpler parts of the 
elementary calculus in the twelfth school year. 

It will now be well to investigate what lies beyond the twelfth 
school year in the field of mathematics. There are two main divi- 
sions in this later field, namely, pure mathematics and applied 
mathematics. Pure mathematics includes such subjects as higher 
calculus, differential equations, functions of a complex variable, 
integral equations, group theory, modern higher algebra, theory 
of numbers, and Einstein's theory of relativity. The first four and 
the last of these subjects make continual use of the calculus. 

Applied mathematics presents a yet wider range of subjects 
which make use of the calculus. Some of these are the mathematics 
of mechanics, of physics, of electricity, of light, of ship-building, 
of statistics, of aviation, of wireless telegraphy, and of engineering. 
Those who are familiar with the courses in engineering know that 
the calculus contributes so much to its theory that much of our 
engineering work would be impossible without the calculus. A 
similar thing may be said of mechanics, physics, and electricity. 
In the last named subject, the calculus is indispensable for a thor- 
ough exposition of the phenomena of alternating currents. 1 A 
knowledge of the calculus is also necessary for an exhaustive study 
of the remaining subjects. Even an elementary textbook on wire- 

1 Charles Proteus Steinmetz, Theory and Calculation of Alternating Current Pheno- 
mena. New York. 1916. 



Important Position Occupied by Calculus 19 

less telegraphy contains some calculus. 2 To-day, the aeroplane is 
built according to the knowledge obtained by the careful methods 
of the laboratory, and the current numbers of aviation journals 
frequently contain articles which cannot be understood without a 
knowledge of the calculus. 3 

The wide application of one of the topics in the calculus is shown 
by the following quotation. "As examples of the utility of these 
theorems (maxima and minima) may be cited the finding of the 
positions and magnitudes of maximum bending moments, of maxi- 
mum stresses, of maximum deflections, of maximum velocities, of 
maximum accelerations of momentum, of the positions of rolling 
load on bridges to give maximum stress in any given member of 
the bridge, etc., etc. All these things are of special importance in 
the practical theory of engineering. In the jointing of pieces in 
machines and static structures, it is never possible to obtain uni- 
form stress over the various important sections of the joint. It is 
of the greatest importance to find the maximum intensities of 
stress on such sections, because the safety of the construction de- 
pends upon the maximum, hardly ever upon the average, stress. 
The average stress on the section is found by dividing the whole 
load on the section by the whole area of the section. Such average 
stresses are often very different from the maximum stress, and no 
reliance ought to be placed upon them as measures of strength and 
safety." 4 

CONCLUSIONS 

Thus it is seen that the calculus is a most important aid in applied 
mathematics and contributes largely to the field of pure math- 
ematics. It is also evident that the elementary calculus is the con- 
necting link between the fields of pure and applied mathematics 
and the school work in the proposed curriculum in mathematics 
preceding the twelfth school year. 

Hence, because of its importance, twelfth year pupils in math- 
ematics should be given an opportunity to become acquainted with 
the elementary calculus, provided it is not too difficult. 

2 S. J. Willis, A Short Course in Elementary Mathematics and Their Application to 
Wireless Telegraphy, London, 1917. 

3 Flight, Sept. 2, 1920, Official organ of the royal aero group of the United Kingdom; 
Aeronautics, Sept. 2, 1920, London; The Aeronautic Journal, Sept., 1920; Aircraft 
Engineering, Jan., 1920, London. 

4 Robert H. Smith, The Calculus for Engineers and Physicists, p. 106, London, 1897. 



IV 

A HISTORICAL SURVEY OF THE NATURAL GROWTH OF 

THE CALCULUS IN THE DEVELOPMENT OF 

MATHEMATICS 

REASONS FOR MAKING THIS HISTORICAL SURVEY 

Different authors of textbooks on the calculus use different 
methods in their presentation of the subject. Not all of these 
methods would prove satisfactory in teaching the elementary 
calculus to senior high-school pupils. Which of these methods may 
prove satisfactory? Or what parts of these methods? Or are there 
additional methods that would prove satisfactory? An aid in 
answering these questions is the light of past experience. According- 
ly, a reason for giving a historical survey of the natural growth 
of the calculus in the development of mathematics is that it may 
serve as a guide in the evaluating of methods of presentation of the 
calculus. 

In many ways the mental development of the child follows the 
mental development of the race. Then a study of the way in which 
the calculus developed in the minds of men throughout the ages 
will suggest methods of how it should now be taught to younger 
pupils. Such a study will point out many of the difficulties that 
arose in the development of the calculus and how these diffi- 
culties were overcome. It will show some methods that are simple 
and satisfactory and others that are cumbersome and un- 
desirable. 

This historical survey will add to the appreciation of the subject 
because it will show that we could not be enjoying the fruits of the 
calculus as the result of its application to the fields of science and 
engineering had it not been for the labors of many mathematicians 
throughout many centuries. This survey will also show that an 
introductory course in the elementary calculus is not too difficult 
for younger pupils. Since the introductory ideas of the calculus 
are within the comprehension of younger pupils, the appreciation 
of the subject is an added reason for introducing the elementary 
calculus into the senior high-school mathematics. 



Historical Survey of Growth of Calculus 2 1 

The chronological order has been followed in this historical sur- 
vey. A representative mathematician has been selected in a con- 
venient span of years, and some of his contributions have been 
discussed in order to show the status of the development of the 
calculus at that particular time. 



THE PERIOD PRECEDING ISAAC BARROW 

The First General Step in the Development of the Calculus. The 
first step in the development of the calculus is found among the 
Greeks, and is known as the method of exhaustion. This method 
was used by Antiphon (c. 430 B.C.) in showing that the area of a 
circle is the limit of the area of an inscribed regular polygon as the 
number of sides of the polygon is increased indefinitely. 1 

Antiphon's first method of procedure was that of inscribing a 
square in a circle. Then he doubled the sides of the square. Then 
he doubled the sides of the resulting polygon, and so on. However, 
Bryson, who lived about this time, was not satisfied with merely 
using an inscribed polygon; he also used a circumscribed polygon. 
His conclusion was that the area of the circle is the limit of the 
area of the variable inscribed polygon and also that of the variable 
circumscribed polygon. 

These illustrations furnish two ideas in the foundation of the 
calculus. One of these is the idea of a limit. The other is the idea 
of the division of a quantity carried on indefinitely. Both these ideas 
are presented to senior high-school pupils in their course in demon- 
strative geometry. Hence, if any of these pupils elect the course 
in the elementary calculus in their last year in the senior high 
school, they will already have become acquainted with the first 
step in the foundation of the calculus. 

A Bit of Integration. There was a near approach to integration 
considered as a process of summation by Archimedes in his method 
for finding the area of a parabolic segment. He made use of the 
summation of a series : A , J4A , {%) 2 A , . . . , and recognized 
that ()4) n approaches zero. 2 

Here, again, the pupils who have taken the work in the math- 
ematics preceding the course in the elementary calculus have already 

^avid Eugene Smith, History of Mathematics, Chap, on The Calculus, Boston, 1921. 
2 Ibid. 



22 Elementary Calculus in Senior High-School Mathematics 

become acquainted with this method in the proof of the following 
proposition in geometry: The area of a parabolic segment made by 
a chord is two-thirds the area of the triangle formed by the chord 
and the tangents drawn through the ends of the chord. 

The Second General Step in the Development of the Calculus. The 
second general step was taken two thousand years later, and is the 
method of infinitesimals. This method was used by both Kepler 
(1616) and Cavalieri (1635). 

Kepler, too, had performed a crude kind of integration (1609). 
He found that "a planet describes equal focal sectors of ellipses in 
equal times." Expressed in our present notation this would be 
So sin k dk = 1 — cos k, where k is the angle of rotation. In the 
wine-year of 161 2, Kepler was confronted with the problem of giving 
practical rules for finding the contents of wine-casks. He thought 
out a method by infinitesimals which he applied in measuring the 
volumes of solids of revolution. He divided areas and volumes 
into elementary parts and then found their sum. Without a fully 
developed theory of infinitesimals, he did this by intuition where 
the theory lacked. His idea of division into elementary parts en- 
abled him to find the form of a cask which would give the maximum 
volume for the material used in its construction. He also knew 
how to test an expression for a maximum value. This test was that 
the difference between the maximum itself and the immediately 
preceding and the immediately following values will be impercep- 
tible if the range covered is made small enough. 3 

This historical sketch thus far shows that the integral calculus 
was developing by itself and not as the inverse of differentiation. 
Later, in discussing methods of presentation of the calculus, the 
problem of the integral calculus will be introduced by itself and 
then it will be shown that integration is the inverse of differen- 
tiation. 

This account of the contribution of Kepler to the calculus shows 
that the idea of infinitesimals may be grasped and used without 
previous training in a rigorous treatment of the theory of infini- 
tesimals. Also, his test for a maximum is both simple and satis- 
factory, and a method similar to it is included in the elementary 
course of the calculus for use in the senior high school. 

1 Heinrich Wieleitner, Geschichle der Mathematik, II Teil, 1 Halfte, p. 107, Leipzig, 
1911. 



Historical Survey of Growth of Calculus 



23 



BARROW, NEWTON, AND LEIBNIZ 

Isaac Barrow. Barrow considered the motion-concept as the 
main idea in his investigations. In one case he derived the path of a 
moving point referred to time and velocity, and in another case he 
derived the velocity of the motion referred to time and path. By 
considering these two ways, Barrow was the first to consider in- 
tegration and differentiation as inverse problems. 

Barrow's differential triangle method will be given here, since the 
method used in the elementary calculus for the senior high school 
for explaining the derivative by means of a geometrical representa- 
tion is based upon it. His method is as follows: 

"Let AP, PM be right Lines given in Position (whereof PM cuts 
the proposed Curve in M,) and let MT touch the Curve in M, and 
cut the right Line AP in the Point 
T. Now to determine the length 
of the right Line PT, I suppose 
the Arch MN of the Curve to be 
indefinitely small, and draw the 
right Lines NQ, NR parallel to 
MP, AP; I call MP, m; PT, t; 
MR, a; NR, e; and grve Names to 
other Lines useful to our purpose, 
determined from the particular 
Nature of the Curve; and then 
compare MR, NR expressed by 

Calculation in an Equation, and by their means MP, PT them- 
selves; observing the following Rules at the same time. 

1. I reject all the Terms in the Calculation, affected with any 
Power of a or e, or with the product of them; for these Terms will 
be equal to nothing. 

2. After the Equation is formed, I reject all the Terms wherein 
are Letters expressing constant or known Quantities; or which are 
not affected with a, or e; for these Terms brought over to one side 
of the Equation will be always equivalent to nothing. 

3. I substitute a for m {MP), and t (PT) for e; by which means 
the Quantity of PT will be found. 

When any indefinitely small Particle of the Curve enters the 
Calculation, I substitute in its stead a Particle of the Curve properly 




24 Elementary Calculus in Senior High-School Mathematics 

taken; or any right Line equal to it, because of the indefinite Small- 
ness of the Part of the Curve." 4 

By means of his differential triangle method, Barrow could find 
the tangent to a curve at any point. He lacked a classified set of 
formulas which would have facilitated this work. Also, there was 
at this time no rigorous treatment of the theory of limits. This 
method of Barrow's justifies the presentation of the derivative as 
the limit of a geometric ratio to pupils who possess only an intuitive 
and elementary knowledge of limits. 

Isaac Newton. The third general step in the development of the 
calculus was the invention of fluxions by Newton (c. 1665). 5 

The method of fluxions played a very important part in the de- 
velopment of the calculus. A little insight into the nature of the 
subject will now be given through a brief explanation of some of the 
terms and notation as used by Newton. 

By a flowing quantity he meant the trace that is being made by 
a moving point. The velocity of the moving point at any particular 
place he called the fluxion. He represented the flowing quantities 
by such letters as x, y, z } and their fluxions by placing a dot above 
the respective letters, as x, y, z. He represented infinitely small in- 
creases or decreases by putting a small zero before the letter. His 
rules for differentiation were similar to the ones used by Barrow. 6 

In our present-day textbooks on the calculus, the topic of rates 
resembles the method of fluxions. It is seen that the method of 
fluxions requires a considerable knowledge of physics. For this 
reason, the elementary calculus based upon infinitesimals and geo- 
metrical representations is more suitable for our American pupils. 

Although the notation of Newton was zealously adhered to by a 
number of the English mathematicians for over a hundred years, it 
finally was replaced by the more convenient notation of Leibniz. 
The Analytical Society in 1820 published two volumes using Leib- 
niz' notation for the calculus ; one was by Peacock and the other by 
Herschel. These volumes were books of examples illustrating the 
new notation. Since then this notation has been used by all ele- 
mentary textbooks. 7 

* Geometrical Lectures by Isaac Barrow, Translated by Edmund Stone, pp. 173, 174, 
London, 1735. These Geometrical Lectures had been published by Barrow in 1670. 

5 David Eugene Smith, History of Mathematics, Chap, on the Calculus, Boston, 192 1. 

6 John Wallis, De Algebra Traclus, pp. 391, 392, 393, Oxonia, Ed. 1693. 

7 W. W. R. Ball, History of Mathematics at Cambridge, p. 122, 1889. 



Historical Survey of Growth of Calculus 25 

Gottfried Wilhelm Leibniz. The following quotation states tersely 
a fundamental difference between the calculus of Leibniz and that 
of Newton. "Leibniz did not reply to this letter [Newton's about 
inverse tangents], until June 21, 1677. In his answer he explains 
his method of drawing the tangents to curves which he says pro- 
ceeded 'not by fluxions of lines but by differences of numbers' ; and 
he introduced his notation of dx and dy for the infinitesimal differ- 
ences between the coordinates of two consecutive points on a curve. 
He also gives a solution to the problem to find a curve whose sub- 
tangent is constant which shows that he could integrate." 8 

Leibniz' idea of an infinitesimal as a minute difference between 
the coordinates of two consecutive points is simpler than Newton's 
fluxion idea because it does not involve the notion of velocity and 
a knowledge of the laws of physics. Because of its greater simplicity, 
the idea of infinitesimal differences is used as the basis of the senior 
high-school calculus. The notation introduced by Leibniz is also 
used in the senior high-school calculus in a somewhat simplified 
form. 

THE EARLY TEXTBOOKS ON THE CALCULUS 

Jean Bernoulli. Although this work of Bernoulli did not appear 
in book form, his manuscripts written in 1691-92 form the first 
integral calculus. 9 

The first lecture concerns the nature and the calculation of the 
integral. His method is that of the inverse of differentiation. He 
says that it is known that dx is the differential of x, and xdx the 
differential of }^x 2 or y 2 x 2 ± a constant quantity, and so on. 
He then generalizes to some extent and forms the following 
table: 

adx is the differential of ax ± a constant 

axdx is the differential of }4ax 2 ± a constant 

ax 2 dx is the differential of y£ax z ± a constant 

ax s dx is the differential of %ax^ ± a constant, and so on. 

8 Ibid., p. 58. 

9 Die Erste Integralrechnung, Eine Auswahl aus Johann Bernoullis Mathematischen 
Vorlesungen iiber die Methode der Integrate und anderes aufgeschrieben zum Gebrauch 
des Herrn Marquis de l'Hospital in den Jahren 1691 und 1692 als der Verfasser sich in 
Paris aufhielt, by Dr. Gerhard Kowalewski, London, 1914. This is the first translation 
of these manuscripts into German. 



26 Elementary Calculus in Senior High-School Mathematics 

From this he deduces the general formula that ax p dx is the differ- 
ential of JL- ,** + !. 

p+l 

Since the pupils in the senior high school are accustomed 
to inductive reasoning, this first attempt at a table of in- 
tegrals is a good method for them to follow in their calculus 
work. 

Bernoulli says that direct, important mathematical problems 
and theorems depend upon the finding of the integral, such as the 
quadrature of surfaces, the rectification of curves, the cubes of 
solids, the inverse tangent method or the finding of the nature of 
the curve from given properties of the tangent, and also such as 
pertain to mechanics, as the method of finding the center of gravity, 
of impulses, of vibratory motion, and so on. Also, through the find- 
ing of the integral we obtain the evolute of curves, and the art 
of finding their nature, and with the help of the evolute of rectifying 
the curves themselves. 

This shows the wide use that was made of the calculus over two 
hundred years ago. Since then its fields of application have been 
greatly extended in pure mathematics, in science, and in engineer- 
ing. This usefulness of the calculus is evidence of its importance in 
the mathematical structure. 

Bernoulli's second lecture pertains to the quadrature of surfaces. 
He says that of the many different parts into which we divide the 
integral calculus, by far the first and most important is that which 
presents the quadrature of surfaces. In accordance with this plan, 
the integral calculus in the elementary course in the senior high 
school is introduced by giving an illustrative problem in finding the 
area of a surface under a parabola. 

Bernoulli considers the surface as divided into an infinite number 
of parts, each of which may be regarded as the differential of the 
surface. If then we have the integral of this differential, that is the 
sum of these parts, then will we also know the desired area. Those 
infinitely little parts of the plane surface can be thought of in 
different ways, according to the most convenient way permitted 
by the details of the plane. Either the plane surfaces would be 
directly divided, as is the custom, by means of parallel lines, as in 
Fig. 2, or by means of an infinite number of straight lines which are 
concurrent, as in Fig. 3, or by means of an infinite number of tan- 



Historical Survey of Growth of Calculus 



27 



gents, as in Fig. 4, or by means of an infinite number of normals to 
the curve, as in Fig. 5. 





Fig. 2 



Fig. 3 





Fig. 4 



Fig. 5 



If the divisions of the surface are parallel, and if x is the abscissa 
and y is the ordinate, the differential of the surface will be ydx; 
namely, the rectangle formed by 
the ordinate and the differential 
of the abscissa. If A C is the given 
curve, then y will have a given 
ratio to x, so that it is expressed 
in x alone. Suppose that A C is a 
parabola, then ax = y 2 or y = 
\/(ax) . The integral of this, which 
is y$x V (ax) or ^3 xy, is the area 
desired. 

This method of Bernoulli's for fig. 6 

finding the area is simple. It is 

also satisfactory, although the idea of limits involved is taken care 
of largely by intuition. This is a justification for the use of this 




28 Elementary Calculus in Senior High-School Mathematics 




Fig. 7 



method somewhat modified in introducing the subject of integra- 
tion as a process of summation in the senior high-school cal- 
culus. 

Marquis de V Hospital. This mathematician wrote the first text- 
book on the differential calculus. The book was published in 1696. 10 

L'Hospital considers a "diff- 
erence" as an infinitely small por- 
tion of a variable quantity that 
increases or decreases contin- 
ually. He gives a number of 
illustrations of differences. Pp 
will be the difference of AP; Rm 
that of PM, Sm that of AM, 
Mm that of the arc AM, the 
triangle MAm that of the seg- 
ment AM, the little space MPpm 
that of the mixtilinear triangle 
AP, PM, and the arc AM. If 
the variable AP is represented 
by x, then dx expresses the value of Pp, and so on. 

This simple method, somewhat modified, is used in the explana- 
tion of the increment of a variable in the senior high-school 
calculus. 

Charles Hayes. Charles Hayes published the first textbook on 
the calculus in English in 1704. 11 

A comparison of different methods used in differentiation at this 
time can be made to some extent by considering the three methods 
that Hayes presents for the differentiation of a product. 

He states the proposition as follows: "If X be multiplied into Z, 
and if the Product be XZ, I desire to know the Fluxion of the Rec- 
tangle XZ; That is, supposing the Sides X and Z to be augmented 
or diminished each by an infinitely little Quantity, I would know 
how much the new Rectangle exceeds or is exceeded by the given 
Rectangle XZ." 

Although Hayes used the fluxional notation, the Leibniz notation 

10 Marquis de l'Hospital, Analyse des infiniment Petits pour V Intelligence des Lignes 
Courbes, Paris, 1696. 

11 Charles Hayes, A Treatise of Fluxions, or an Introduction to Mathematical Philoso- 
phy, Gent., London, 1704. 



Historical Survey of Growth of Calculus 



29 



will be used here in the methods he gives since it is more familiar 
to the present-day reader. 

First method. A geometrical figure is used. If X be multiplied 
by Z, the rectangle is XZ, and we suppose that half the little in- 
crement of X to be }4dx, and 
half the little increment of Z to idx 
be }4dz; it is evident that the idx 
differential of the rectangle is 

X-y 2 dz + Z-y 2 dx -f-r. x 

Again, suppose that half the 
infinitely little decrement of X and 
of Z to be ydx and y 2 dz respec- 
tively, then the decrement of the 'dz-gdz 
rectangle XZ is equal to X-ydz FlGt 8 
+ Z-y 2 dx — r, and adding the 

increment and the decrement into one sum, we have Xdz -f Zdx for 
the differential of the rectangle XZ. 

The second method. This method makes use of the algebraic 
process. The "Incomparable Mr. Newton" shows how to find the 
differential of any rectangle in a manner nothing differing from this, 
only expressed another way. For instance, to find the differential 
of the rectangle XZ, he supposes the differential of X and of Z to be 
dx and dz. Then, 



X+}4dx 

z+y 2 dz 
xz+y 2 dx-z 

+ ydzX+ydx-dz 



x-y 2 dx 

z-y 2 dz 

xz-y 2 dx-z 

-ydzX+ydx-dz 



XZ + y 2 dx Z + y 2 dz X + ydx -dz XZ - y 2 dx Z - y 2 dz X + %dx 'dz 



By subtracting these two results, he gets dx'Z + dzX as the differ- 
ential of the rectangle XZ. 

A third method. The differentials of the sides X and Z are dx 
and dz, and therefore the sides of the rectangle become X + dx 
and Z + dz, and the rectangle itself is XZ + dx'Z + dz'Z + dz'dx. 
By subtracting the given rectangle XZ from this result, the re- 
mainder dx-Z -f dzX gives the differential of the rectangle XZ (the 
term dz'dx being infinitely small in comparison of either of 
these) . 



30 Elementary Calculus in Senior High-School Mathematics 

These three methods suggest a satisfactory way of presenting 
the steps used in obtaining the fundamental rule for differentiation 
to a class of beginners. This suggested method, which is used in the 
senior high-school calculus, first presents the derivative as the 
limit of a geometric ratio. Then there is obtained directly from 
this geometric representation the algebraic representation of the 
derivative as the limit of a fraction. Then from an analysis of this 
last form, the four steps of the fundamental rule are obtained di- 
rectly. 

Christian Wolf. Wolf also used a geometric method for finding 
the differential of a product. 12 

His supposition is that if two quantities are multiplied together 
as xy, then the differential of one factor multiplied by the other 
factor for each of the factors and the sum of these two products, 
which is xdy + ydx, will be the differential desired. That is, d (xy) 
= xdy + ydx. 

In this demonstration, xy is represented by the rectangle CABD 
whose base AC = x, and altitude DC — y. If we suppose the sides 

to be increased, then CA changes 
. H into CL = x + dx, and CD into 

^ p- CE = y -f- dy, and the rectangle 



CABD changes into CLGE. 

Thus, the differential of xy 
itself is the difference between 
the rectangles CABD and CLGE. 
Whereby, d (xy) is found from xy 
+ ydx + xdy + dxdy — xy = ydx 
u D a + xdy + dxdy; namely, ALHB -f 

fig. 9 DBFE + BHGF. Then in the 

rectangle ALHB = ydx, AL = dx 
is a finite quantity; and HGFB = dxdy will be its difference. But 
this is also the difference in the rectangle DEFB. Accordingly, since 
HBFG = dxdy is the difference of each of the two rectangles ALHB 
and DEFB, its relation to these two rectangles is cancelled; conse- 
quently, the difference between the rectangle CABD and CLGE, or 
the differential of xy itself, is ydx + xdy. 

Although Wolf's method is an improvement in the method for 

12 Christian Wolf, Rlementa Matheseos Universae, Second Edition, V. I, p. 545, 
Magdeburg, 1730. 



Historical Survey of Growth of Calculus 3 1 

finding the differential of a product it was not until Cauchy gave 
a rigorous demonstration of his method of limits that a completely 
satisfactory form was obtained. Cauchy's method will be com- 
mented upon later. 

Wolf extends the content of the formula d (xy) = xdy + ydx 
to include the cases of the product of several variables and also of 
the powers of a single variable by means of a substitution method. 
Suppose that the expression to be differentiated is the product of 
three variables vxy. Let vx = t, then vxy = ty, and consequently, 
d (vxy) = tdy -f- ydt by use of the formula, d (xy) = ydx + xdy. 
Then for t substitute its value vx, and we have d (vxy) = vxdy + 
vydx + xydv. 

This method could probably be used to advantage in a calculus 
course in senior high-school mathematics. 

Leonhard Euler. Euler was the first to introduce the symbol A x 
for the difference or increment of x. lz 

M. Cousin. Cousin was a strong advocate of the method of 
limits. He uses two propositions from geometry as illustrations of 
the method of limits. One of these was first used by Bryson, and 
has already been mentioned. It is the proposition that states that 
the circumscribed and inscribed polygons approach the circle as 
their limit. The other is the proposition concerning the ratio of the 
area of the ellipse to that of the circle described upon its major axis 
as a diameter. 14 

Both these propositions are contained in the geometry of the 
senior high school, and hence are within the mental grasp of the 
pupils of the senior high school. 

Lazare Carnot. Carnot gives a rigorous discussion on the proper 
use of infinitesimals and of limits. He shows that the method of 
infinitesimals is imperfect but satisfactory and that the method of 
limits is both exact and satisfactory. 15 

Augustin Louis Cauchy. Cauchy gives a rigorous proof of the 
method of limits as used in finding the derivative. 16 

13 Moritz Cantor, Geschichte der Mathematik, V. 3, p. 725, Leipzig, 1898. This page 
is a part of Cantor's discussion on Euler's Differentialrechnung of 1755. 

14 M. Cousin, Lecons de Calcul differ entiel et de Calcul integral, V. I, pp. x-xiv, 
1777. 

16 Lazare Carnot, Reflexions sur la Metaphysique du Calcul infinitesimal, Paris, 
1797. 

16 M. Augustin Cauchy, Lecons sur le Calcul differentiel, p. 17, Paris, 1829. 



32 Elementary Calculus in Senior High-School Mathematics 

He shows that A y = / (x + i) —/(#), where A x = », an in- 

finitesimal. He then writes — - = — — and says that 

A # « 

as the two terms A x and A y approach zero as a limit, the value 
of this same expression will approach another limit, either positive 
or negative; provided, of course, a limit exists. This limit deter- 
mines a value for each particular value of x, but varies also with x. 
Thus, for example, if/ (x) = x m , where m is an integer; then the 

• / (x + i) ~ f (x) , (* + *) w - (x) m _. . .„ 

expression^ — ! — £ ^—'becomes- — ' — ^— . This will 

i i 

give rax™ ~ l -f- — — x m ~ 2 i + . . . + i m ~ l , by means of 

1.2 

the binomial theorem and the combining of terms and the division 

by the denominator i. This expression then has for its limit, as i 

approaches zero, the quantity mx m ~ .*, a new function of x. The 

same will be true in general ; only, that the form of the new function 

f (x + i) — f (x) 
which is the limit of the expression — J depends for 

i 
its form upon the given function y = f (x). Hence, in order to indi- 
cate this dependence, the new function is given the name "derived 
function" or derivative. Its symbol is/' (x). 

Now that the method of obtaining the derivative by the use to 
limits as described here has been rigorously demonstrated by Cauchy, 
this method may be used in future work without again proving it. 

CONCLUSIONS 

This historical survey furnishes some valuable suggestions as 
to the way the elementary calculus should be presented to a class of 
senior high-school pupils. There has been much use made of geo- 
metric representations. Frequently these have then been expressed 
in algebraic form. Then from an analysis of these algebraic forms, 
the working rule or the principle of the process has been obtained. 

This survey also shows that mathematical intuition has played 
an important part in the development of the calculus. Mathematical 
intuition made possible methods that, although not exact, were 
satisfactory. 

The illustrations and the discussions of the topics in this historical 
survey show that the introductory ideas of the elementary calculus 
are not too difficult for pupils in mathematics in the last year in the 
senior high school. 



COMPARISON OF TEXTBOOKS ON THE ELEMENTARY 
CALCULUS FOR BEGINNERS AND FOR SELF- 
INSTRUCTION 

THE STANDARD METHOD 

A thorough investigation has already been made as to the nature 
of that part of the calculus suitable to our high schools. The con- 
clusion is that "The work in calculus should be largely graphical 
and closely related to that in physics; the necessary technique 
should be reduced to a minimum by basing it wholly on algebraic 
polynomials. No formal study of analytic geometry is presupposed 
beyond the plotting of simple graphs." l 

With this assumption, it would be well, in the examination of 
textbooks, to notice how they contribute to the standard thus set 
forth. 

Furthermore, it is desirable to consider the method of approach 
to the subject which characterizes these various works. For ex- 
ample, does the author use the method of infinitesimals or does he 
favor the method of limits? Is the treatment made from the 
scientific viewpoint or from that of the learner's ability to grasp 
the subject? If the psychological method is followed, a work in- 
tended for younger pupils would naturally be genetic in character, 
being based upon the historical steps by which the calculus de- 
veloped in the mind of the race. 

The scientific viewpoint is illustrated in the older form of text- 
books on high-school algebra. Many definitions are first given, 
many abstract ideas are explained in detail, and a number of topics 
of little value to the learner such as the extraction of the roots of 
polynomials are treated at length. The scientific method is the 
proper method for mature minds with sufficient preparatory 
training in mathematics but, as experience in teaching algebra has 
shown, it is not the method best suited to the minds of younger 
pupils. 

1 Elective Courses in Mathematics, fourth revision, a preliminary report by The 
National Committee on Mathematical Requirements, March, 192 1. 



34 Elementary Calculus in Senior High-School Mathematics 

TEXTBOOKS REVIEWED 

American. There are almost no American textbooks of this 
type. 

Elementary Analysis, by W. A. Granville and P. F. Smith (Bos- 
ton, 1 910). The first part of the book is devoted to analytic geom- 
etry and the second part to the elementary calculus. The object of 
the book is to prepare a simple and direct exposition of those por- 
tions of mathematics beyond trigonometry which are of importance 
to students of natural science. There is an intentional avoidance of 
anticipatory difficulties. Processes which are natural are intro- 
duced without explanation. Much use is made of graphical repre- 
sentation. 

Calculus with Applications, an Introduction to the Mathematical 
Treatment of Science, by Ellen Hayes (Boston, 1900). Its aim is two- 
fold: for purposes of culture and for the purpose of learning, in as 
simple a way as possible, what the calculus is and what it is for. 
The book is a reading lesson in mathematics and the examples are 
taken from mechanics and astronomy. Relationship is expressed as 
"a quantitive change in the cause which is accompanied by a 
quantitative change in the effect." The term function is applied to 
the quantity which necessarily changes because of a change in a 
variable with which it is connected. The chief object of the book is 
to show how to use the operations of the calculus. 

Elementary Calculus, by William F. Osgood (New York, 192 1). 
This book, although intended for college use, is included in this 
review because it is probably the latest textbook on this subject. 
"The object of this book is to present the elements of the Differen- 
tial Calculus in a form easily accessible for the undergraduate." 
Professor Osgood emphasizes the use of graphical methods. 

English. There are a number of English textbooks on the 
calculus for beginners in which the authors have simplified the 
treatment of the subject. 

The Calculus for Beginners, by W. M. Baker (London, 1919), 
"The differentiation and integration of the simpler standard forms 
are applied as early as possible to the determination of maxima and 
minima, of the areas and lengths of curves, of volumes of revolution, 
to the solution of problems in mechanics and physics and to the 
expansion of simple trigonometric functions." The author treats 



Comparison of Textbooks on Elementary Calculus 35 

the method of limits at considerable length. Graphic illustrations 
are frequent. The book contains 750 examples largely of an applied 
character. 

Differential Calculus for Beginners, together with a second book, 
Integral Calculus for Beginners, by Alfred Lodge (London, 1913). 
The object of the differential calculus is to provide an easy introduc- 
tion to the subject for those students who have to use it in their 
practical work, by making them familiar with its ideas and methods 
within a limited range. "Stress is laid at the outset on the graphic 
representation of functions, as by this means a vividness and 
reality is given to the processes of the calculus which is not easily 
obtained without such ocular assistance. In this way the theory of 
maxima and minima becomes very simple and interesting." The 
author introduces the differential before the derivative and gives 
an excellent discussion of the subject. He introduces the differential 
coefficient or derivative by simply stating that it is the ratio dy / dx 
when y = f (x). His integral calculus is written for students of 
physics and mechanics and contains many applied problems. 

An Introduction to the Differential and Integral Calculus for the 
Use of Engineering and Technical Students, by John Stoney (Lon- 
don: no year). 2 The practical side of the subject is dealt with. 
"When two quantities are so connected that a definite value given to 
the one produces a corresponding definite value in the other, each is 
said to be a function of the other." The author introduces the 
derivative as a rate. Throughout the book the work of the calculus 
is applied without any attempt at an exhaustive and rigorous treat- 
ment of the theory. However, he frequently introduces abstracts of 
such a treatment. This makes the work lack unity. The author 
includes such subjects as partial differentiation and hyperbolic 
functions. The pupils would require a knowledge of physics and 
engineering. 

The Elements of the Differential Calculus, by W. S. B. Wool- 
house (London, 1852). Woolhouse uses the method of infinitesimals, 
and says that this method is just as rigorous as that of limits. "In 
comparing the relative values of any two infinitesimals, the rejec- 
tion of terms involving infinitesimals of higher orders is, in effect, 
precisely the same as that of proceding to the ultimate ratio of the 
infinitesimal quantities by the method of limits and such rejection 

2 The Library date is 1920. 



36 Elementary Calculus in Senior High-School Mathematics 

of values may be said to be the operation of cropping down the 
quantities to their ultimate or limiting relative positions." 

Easy Lessons in the Differential Calculus; Indicating from the out- 
set the utility of the processes called differentiation and integration, by 
Richard A. Proctor (London, 1887). Proctor approaches the 
subject through the laws of falling bodies, and spends several pages 
on this discussion. He treats the integral calculus simply as a part 
of the differential calculus, and considers "the differential calculus 
as the science which deals with the rate at which variable quanti- 
ties increase or diminish." 

Calculus Made Easy, by S. P. Thompson (London, 1919). The 
book is written in very simple and interesting language and shows 
how easy some of the apparently difficult things in calculus may be 
made. The author first explains in unscientific language the mean- 
ing of the symbols for differentiation and integration. He discusses 
each new topic at considerable length, and often explains the dif- 
ficulties in near-slang language. He makes considerable use of 
graphical representation. "The aim of this book is to enable begin- 
ners to learn its language, to acquire familiarity with its endearing 
simplicities, and to grasp its powerful methods of solving problems, 
without being compelled to toil through the intricate out-of-the-way 
(and mostly irrelevant) mathematical gymnastics so dear to the 
unpractical mathematician." 

French. There are no separate French textbooks on the ele- 
mentary calculus for beginners since this kind of work is included 
in their courses in algebra. 

Precis d Algebre, by Carlo Bourlet, Paris, 1909. This book in- 
cludes the work on the calculus as prescribed in Plan d Etudes et 
Programmes d ' Enseignement dans les Lycees, 1 907-1 908. The outline 
of this work is given in Section I of this dissertation under the Lycees 
(France). 

Algebre, Second Cycle, by Emile Borel, 3rd ed., (Paris, 1905). 
This book, also, was written to meet the requirements as set down 
by Programmes Officiels de 1905. This work for the Classe de Seconde 
(Sections C and D) in Algebre, so far as it relates to the simple 
calculus, has already been indicated in Section I of this dissertation 
under the Lycees (France). 

Cours d'Algebre Elementaire Conforme aux Dernier s Programmes de 
V Enseignement Secondaire (1902), by F. G.-M. (Paris, 1909). As the 



Comparison of Textbooks on Elementary Calculus 37 

title indicates, this book also gives the subject-matter as required 
by the Programmes Officiels. However, the subject-matter is more 
comprehensive than in the two preceding books, and there are 
also a greater number of applications of the derivative. The graph 
work also is discussed at greater length than in the preceding books. 

Cours de Maihematique a V Usage des Candidats a VEcole Polytech- 
nique et Autres, par Charles De Comberousse, Tome Troisieme 
(Paris, 1887). The scientific method is used. The author gives a 
good classification of simpler functions into algebraic and tran- 
scendental functions and their subdivisions. 

German. There are a number of German textbooks on the 
calculus written for beginners and for self-instruction. 

Elementarbuch der Differential- und Integralrechnung mit Zahl- 
reichen Anwendungen aus der Analysis, Geometrie, Mechanik, und 
Physik, fur hohere Lehreanstalten und den Selbstunterricht, von Dr. 
Alfred Donaldt (Leipzig, 1901). The first part consists of an intui- 
tive introduction into the calculus by means of simple differentiation 
and integration. Then the scientific method is used in separate 
treatments of differentiation and integration. 

Elemente der Differential und Integral Rechnung, by Ludwig 
Tesar (Leipzig, 1906). This book is probably within the intellectual 
capacity of our senior high-school pupils. Asymptotic curves are 
used to prepare the learner's mind for the work in limits. He intro- 
duces the differential coefficient by means of a time-speed table. 
Integration is introduced as a summation process. 3 

Die Anfangsgriinde der Differ entialrechnung und Integralrechnung, 
by Dr. Richard Schroeder (Leipzig, 1905). This book is intended 
for self-instruction and for use in the higher schools. However, it 
does not differ materially from the traditional calculus texts for use 
in colleges. It was written to meet the needs of the new course of 
study to be introduced into the Prussian secondary schools. The 
book would be unusable in our senior high school. 3 

CONCLUSIONS 

This review of textbooks shows the tendency to put the study of 
the elementary calculus earlier in the mathematics course. It also 
shows that textbooks on the elementary calculus are being written 

8 These two books were reviewed by Mr. M. A. Norgaard. 



38 Elementary Calculus in Senior High-School Mathematics 

now from the standpoint of the learner's ability to grasp the subject. 
The English textbooks emphasize graphical representation and 
the abundant use of material from physics and mechanics. Their 
graphical representation necessarily requires the use of the algebraic 
polynomial. Such books are the most hopeful of any we have in 
teaching the calculus to younger pupils. 

A study of these works would seem to show that a course in 
the elementary calculus in the senior high school should have its 
theory based upon algebraic polynomials with the explanations given 
by means of graphical representations. The work should be closely 
allied to physics and mechanics by the use of a sufficient number of 
suitable problems from these fields to give reality to the subject 
and an insight into the wide use of the calculus in applied fields. 4 

4 This is the method that is followed in the course of which a synopsis is given in 
Section VII of this study. 



VI 
THE TREND OF AMERICAN EDUCATION IN GENERAL 

THE INCREASED LENGTH OF SCHOOL ATTENDANCE 

Not many years ago the educational scheme for our public schools 
was extremely limited. A few years of a few months each was the 
extent of the intellectual training of a pupil, and this was looked 
upon as sufficient for the conditions of those times. As conditions 
changed, so also did the scheme of education, but less rapidly. The 
reason for this is that our educational system, like the church in this 
respect, is among our more conservative institutions. 

Before the beginning of our present attempt at a readjustment in 
our educational system of which the initial wave is the junior high- 
school movement, the type child attended school eight years of nine 
months each. There were some children who began in the kinder- 
garten and completed a four-year high-school course. The educa- 
tional scheme for these children covered from twelve to fourteen 
years. To-day there are indications that the period of compulsory 
education of a certain sort should be extended to the age of eighteen 
years, such as part-time education for young people who leave school 
at the age of fourteen to become wage-earners. Also, the bright pupil 
who is without the necessary financial means may be able in the 
future to extend his number of years in study by the aid of a govern- 
ment subsidy. 1 There is an inclination, too, on the part of the Labor 
Party to favor an extension of the age limit of compulsory educa- 
tion in order that its members may enjoy greater security against 
the encroachments of child labor. 2 

It is also true that many young people continue their studies for 
many years after they have entered the work-a-day world. The 
discussion thus far shows that there is a marked tendency to render 
indistinct the dividing line between the period of formal education 
as represented by our public school system and the time when the 
individual ceases educational activity. This raises the question, 

1 Notes on lectures by Professor S. S. Colvin, Teachers College, summer, 1918. 

2 Notes on lectures by Professor D. E. Smith, Teachers College, winter, 1920. 



40 Elementary Calculus in Senior High-School Mathematics 

is there a limit as to the number of years one should indulge in 
study, especially in the study pertaining to a special field? 



THE CHANGED POINT OF VIEW IN EDUCATION 

We hear this question, is not the time spent in our present 
scheme of education already too long? Or, does it not take too 
many years out of the lives of the children in preparation for life? 
This latter question must be answered in the affirmative, if in think- 
ing of education as a preparation for life we consider that the pupil 
should be more or less isolated from the ordinary interests of life 
during this period and that he should be imbibing theory which in 
later years he shall put into practice in meeting life's situations. 
There is too much indefiniteness and uncertainty of attainment of 
the desired end in thus looking at education as a preparation for 
life itself. 

However, the scheme of education resulting from our present 
attempt at readjustment in our educational system considers edu- 
cation not merely as a preparation for life but as life itself. By the 
statement that education is life itself, among other things, is meant 
that the pupil is forming habits that he will desire to maintain and 
strengthen throughout his lifetime, that the methods of obtaining 
knowledge that he is acquiring are the kind of methods that he will 
use for this purpose throughout his lifetime, that the kind of think- 
ing in which he is being trained is the kind of thinking that will 
serve him best in meeting other situations in life, that the ideals 
he is forming will be the nucleus for the development of those more 
mature standards which will serve as the stabilizers and guides of 
his life, and that the attitudes that he is being led to assume are 
such that they will need no radical change but merely accentuation 
to make him an acceptable member of human society. Of course, 
this new way of viewing education carries over much that has al- 
ways been in education, but this changed point of view is highly 
important in that it makes for a better selection of the material 
that should be included in our school work and of the methods in gen- 
eral in secondary education. This changed point of view means that 
educational activity is a constituent, not an incidental, part of life 
itself, and that among the more intelligent individuals, and especial- 
ly the leaders of human society, educational activity never ceases. 



Trend of American Education 41 

THE VIEWS OF OTHERS 

Professor W. H. Kilpatrick considers that the typical unit of 
educational procedure should be "wholehearted purposeful activity 
proceeding in a social environment, or more briefly, in the unit 
element of such activity, the hearty, purposeful act." 3 

Professor David Snedden says, "In a sense any concrete job 
undertaken in a vocational school where the realization of valuable 
results in the product constitutes an important end, might be called 
a 'project,' but to be an 'educational project' such a job . . . must 
be of such a nature as to offer large opportunity, not only for the 
acquisition of new skill and experience in practical manipulation, 
but also for applications of old and learning new 'related knowledge', 
art, science, mathematics, administration, hygiene, social science, 
etc." 3 

Professor E. L. Thorndike expresses a more general view of edu- 
cation. He says, "The arts and sciences serve human welfare by 
helping man to change the world, including man himself, for the 
better. The word education refers especially to those elements of 
science and art which are concerned with changes in man himself." 4 

One never gets too old to study provided he has had sufficient 
earlier training in habits of mental activity. That an individual 
who continues to develop and strengthen his mental habits until 
he has reached the age of twenty-five can, if he wishes to, assimilate 
new thought for the remainder of his lifetime, is a frequent interpre- 
tation of a discussion on habit by a world-famous psychologist. 5 

The Need of More Trained Mathematical Minds and the Source of 
Supply. Since America has become the greatest manufacturing 
and engineering nation in the world, her need of more men with a 
wider knowledge of mathematics is rapidly making itself apparent. 
Hence, the elementary calculus would be the logical subject to 
supply the want of more vigorous and profitable mental food for the 
bright pupils of our senior high school who are mathematically in- 
clined. That our bright pupils are in need of more vigorous mental 
food has been confirmed by investigation. If we divide pupils into 
three classes, bright, ordinary, and dull, and compare their educa- 

3 Notes on lectures by Professor William C. Bagley, Teachers College, summer, 1920. 

4 E. L. Thorndike, Educational Psychology, V. I, p. i, New York, 1919. 

5 William James, Principles of Psychology, V. I, pp. 121, 122. 



42 Elementary Calculus in Senior High- School Mathematics 

tional attainments with their general intelligence ability, it will be 
found that the last two classes have reached a standard for their ed- 
ucational attainments that is as high as their general intelligence 
ability will warrant, but that the bright pupils fall below in their 
standard of educational attainments as compared with their general 
intelligence ability. This waste of fertile mental resources is not 
only an individual loss but also a national one, and should be 
promptly eliminated. 6 

The following quotation gives the modern point of view in edu- 
cation, "We are attempting to use every bit of native ability of the 
pupil and by so doing eliminate the waste of intelligence that 
is now going on in our schools to-day. The wasted intelligence 
is much greater in amount than most of us have realized. Ever 
since the introduction of mental and educational tests we have 
all been aware of this wastage, but we have not been able to mea- 
sure it accurately. The combination of mental and educational 
tests will enable us to do so. It will show us just where — upon 
what school or class or pupil — pressure must be brought to bear 
in order to eliminate the waste and make the school, class, or 
pupil work up to the extent of the native capacity. 

"As a matter of fact we shall see, what we long suspected and 
what we have only recently consciously realized, that the greatest 
amount of waste exists among the brighter pupils in a class or 
among the better schools in a school system. It is as a rule the 
more intelligent pupils that are working below capacity, even 
although they are keeping well up to the average level of the group. 
We have been pushing and cramming the duller children while 
the brighter ones have been allowed to loaf. The bright child 
is the most retarded child in our schools. The dull child is the most 
accelerated. The bright child is the laziest child, and the dull 
child is the most industrious. We are now ready by means of 
combined mental and educational tests to equalize the pressure, 
and stimulate both the bright and the dull so that they may work 
up to their respective capacities. If we can accomplish this, we 
shall have happier and better children. Lack of adequate stimulus 
undoubtedly leads to much disciplinary trouble in our schools. 
Habits of mental laziness acquired in school often persist through 
life, and there are undoubtedly many adults at the present time, 

6 Notes on a lecture by Dr. Rudolph Pintner, summer, 1920. 



Trend of American Education 43 

who have splendid native ability and who do not know it because 
the school has taught them to be satisfied with a mediocre type of 
accomplishment." 7 

CONCLUSIONS 

It has been shown that the bright pupils in our high school 
need more vigorous food. A course in the elementary calculus 
will satisfy this need for those who are mathematically inclined. 

It has been shown that many individuals study along certain 
lines the greater part of their lifetime. The person who may 
study along the line of applied mathematics will need a knowledge 
of the two operations, differentiation and integration, in order to 
understand some of the most valuable mathematical contributions. 
Such a person would have received in a course in the elementary 
calculus in the senior high school sufficient instruction in differ- 
entiation and integration to continue further his study of the 
calculus without the aid of a teacher. 

7 Rudolph Pintner and Helen Marshall, "A Combined Mental-Educational Survey," 
Journal of Educational Psychology, Jan., 1921, p. 32. 



VII 

SUGGESTIONS FOR A MODERN PRESENTATION OF THE 
ELEMENTARY CALCULUS 

THE AIM OF THESE SUGGESTIONS 

These suggestions are intended to show how to simplify the 
subject-matter of the elementary calculus and its presentation, 
how to make the topics definite, and how to lead the pupils to 
see the theory as a related whole. A feature of this high-school 
course in the elementary calculus which tends to make the work 
definite is that the general purpose of the course harmonizes with 
the general purposes of education. One of these general purposes 
of education is to impart to the coming generation the knowledge 
which has become the heritage of the race. 1 It has been shown 
that the calculus is a part of this heritage since it is the product 
of the minds of many men throughout many years. In the selec- 
tion of the material for this course and in the manner of the presen- 
tation of this material, the particular aim of the course has been kept 
constantly in mind. 2 This is a feature that contributes materially 
to the definiteness of the work. The unity of the theory has been 
made to stand out more clearly by the use of the same kind of 
material in nearly all of the explanations. The presentation of 
the subject-matter has been simplified by making use of material 
that lies within the experience of the pupil. This experience may be 
real or imaginary. 

THE DIVISIONS OF THE DISCUSSION 

The first part of these suggestions deals with the plan of the 
course, the aim, and the method of presentation. The second 
part deals with the particular topics included in the course. 
This second part is written in the form of a synopsis of a text- 
book on the elementary calculus suitable for use in our American 
high schools. 

1 Notes on lectures on the Technique of Teaching, by Professor W. C. Bagley, 
summer, 1920. 

2 See statement of the particular aim a few paragraphs later under the heading 
"The Aim of the Course," p. 48. 



Suggestions for Presentation of Calculus 45 

First Part 
general features influencing the plan of the course 

The Viewpoint. There are a number of textbooks on the calculus 
for beginners and for self-instruction. Many of these textbooks 
were written by mathematicians for use in schools abroad. With 
probably one or two exceptions they are not satisfactory for use 
in our senior high school because our schools differ from those 
abroad both in the character of the pupils and in the nature of the 
organization. The pupils in our senior high school come from a 
greater variety of walks in life than is the case in Europe, and 
although they probably possess greater native ability this ability 
is less highly trained. Also, in our high schools there is little offered 
in the curriculum that leads definitely to the strong technical 
courses of more advanced work as is the case in schools abroad. 
Hence it is necessary that we consider the plan of a course in the 
elementary calculus from the point of view of conditions in our 
secondary schools and especially as these relate to the mental 
needs and capacities of the pupil. 

Important points in estimating the status of a course or subject 
of study in our secondary schools are use, appreciation, mental 
discipline, interest, and degree of difficulty. These features will now 
be discussed. 

Use. Without the calculus much of the work in engineering 
would be impossible, a large part of the work in science would be 
crippled, and a large field of development in pure mathematics 
would be eliminated. However, an understanding of the work in 
any one of these fields must go hand in hand with a knowledge of 
the calculus if a person wishes to apply the latter in such a field. 
Hence the skill and knowledge imparted in a calculus course in the 
senior high school does not equip the pupil to make any worth-while 
use of his knowledge of the subject in applying it to real conditions 
within his experience at the time, but it does enable him to use the 
calculus that he will find in the field in which he intends to specialize 
later. This preliminary course also tends to build up that hierarchy 
of habits which is necessary to a satisfactory understanding of the 
more advanced work in the calculus. 

In the realm of pure mathematics the vast fields of function 
theory, integral equations, and differential equations make con- 



46 Elementary Calculus in Senior High-School Mathematics 

tinual use of the calculus. Indeed, it would be a great advantage to 
the investigator in these fields if his calculus had grown up with 
him in the way that his geometry, algebra, and trigonometry have 
done. 

The calculus unifies the related mathematics that has preceded it 
since it makes use of arithmetic, geometric, algebraic, and trigo- 
nometric ideas indiscriminately as occasion demands. It also shows 
the pupils the use of many of the things that they have studied in 
the preceding branches of mathematics and thus serves especially 
well as a practical review of the parts of the preceding mathematics 
that are worth remembering. The calculus is also found useful to 
the average well-informed citizen in his general reading of scientific 
articles of a popular nature. 3 

Appreciation. A course in the elementary calculus in the high 
school will add to the pupil's appreciation of mathematics since it 
will open up to him vistas of the wide application of mathematics. 
This course also acquaints the pupil with two new operations, dif- 
ferentiation and integration. These two tools are so powerful that 
the pupil cannot help but marvel at the far-reaching influences of 
the principles of mathematics. 

The appreciation of a subject is a point worthy of greater con- 
sideration than may at first be apparent. Since the appreciation of 
the subject contributes toward the cultural development of the 
learner, it also contributes toward the general development of cul- 
ture in our country as a whole. This is an important matter, since, 
because we have become the richest country in the world, it should 
not only be our desire but it is also our obligation to strive toward 
a high standard of culture. 

Disciplinary Value. Educational psychology has shown the 
fallacy of the view "that one of the chief values of education is to be 
found in the cultivation of such mythical powers of the mind as 
reason, attention, imagination, discrimination, and the like." 4 
That is, even if it is conceded that a subject has disciplinary value, 
this reason alone would not be sufficient for including it in the 
curriculum. However, if the subject has disciplinary value of a 

3 The Amount of Mathematics the General Reader should Know as Found by Scan- 
ning the Articles in a Standard Encyclopedia: a class investigation conducted under 
the direction of Professor D. E. Smith, winter, 1920. 

4 S. S. Colvin, An Introduction to High School Teaching, New York, 1918. 



Suggestions for Presentation of Calculus 47 

particular nature that is highly desirable, then its disciplinary value 
would be a contributory reason for including the subject in the 
curriculum. 

In the case of the elementary calculus, an outstanding feature is 
the specific training that the learner receives in the function con- 
cept. There is no attempt made to develop the learner's ability to 
recognize the underlying function relations of the conditions under 
consideration by means of a mere presentation of the theory of the 
calculus given in abstract form. This ability is developed largely, to 
a greater or less extent, unconsciously in the learner's efforts to 
apply the theory to the material taken from the fields of science, 
engineering, physics, mechanics, aviation, wireless telegraphy, and 
the like, and to the simpler material given earlier in the course which 
has been suitably modified so that the pupil will not go far astray in 
the learning process. 

In brief, since the calculus is the mathematics of nature and treats 
of small changes in related conditions, it develops the kind of 
thought that everyone makes use of in considering the problems 
that come up in our daily lives. In fact, one college professor of 
mathematics has stated that the elementary calculus consists of 
ninety-nine per cent common sense and one per cent calculus. 5 

Interest. There should be little attempt to arouse interest arti- 
ficially in the elementary calculus by means of far-fetched problems 
from specialized fields. Indeed there need be no resort to artificial 
means to stimulate the learner to an interest in the subject, since he 
has chosen the elective courses in the mathematics of the senior high 
school because he likes mathematics or because he intends to follow 
lines in which he will have further need of the science. Nevertheless, 
in the exercises of the course in the elementary calculus there are 
included many applied problems that will contribute toward stimu- 
lating an interest in the subject. 

Degree of Difficulty. By degree of difficulty is meant, is the sub- 
ject within the mental grasp of the pupils for whom it is intended? 
Since the elementary calculus is taught in many schools abroad to 
pupils of approximately the same age as our pupils in the twelfth 
school year, or the last year of the senior high school, it surely is not 
too difficult for our selected pupils whose tastes have led them to 

5 Notes on lectures by Doctor H. F. Stecker, The Pennsylvania State College, 
winter, 1910. 



48 Elementary Calculus in Senior High-School Mathematics 

elect to study the subject. In fact, this introductory course in the 
elementary calculus does not contain work as difficult as that 
required to solve many of the puzzle problems and problems on 
obsolete subjects in the arithmetic of bygone days. Furthermore, a 
historical survey of the development of the calculus shows that the 
more elementary topics lie easily within the grasp of pupils in the 
last year of the senior high school. 

Hence, the course in the elementary calculus is useful, it generates 
an appreciation of mathematics, it develops the kind of thought 
that can be applied to other situations in life, it is interesting to the 
pupils who elect mathematics, and it is not too difficult for such 
pupils. 

THE AIM OF THE COURSE IN ELEMENTARY CALCULUS IN THE SENIOR 
HIGH-SCHOOL MATHEMATICS 

The aim of the course is twofold: to give to the pupils the be- 
ginning of the mathematical language they will have to use if they 
continue further their studies in mathematics; and to give to pupils 
such an introduction into the subject as will enable them to pursue 
its study further without the aid of a teacher. 

In working to attain this aim there will be built up, as has already 
been stated, that hierarchy of habits which is necessary to a satis- 
factory understanding of the more advanced calculus. Also, this 
course in the elementary calculus will be found to contribute to the 
satisfaction of the innate desire of such pupils who display a love for 
mathematics. 

THE METHOD OF PRESENTATION 

The method of presentation is molded largely according to the 
suggestions derived from a historical survey of the development of 
the calculus in the mind of the race. This development was simple 
and natural. Since the development of the individual mind is to 
some extent analogous to the development of the mind of the race, 
many of these primitive methods will be found to be well worth 
following in teaching the elementary calculus to high-school pupils. 

The suggestions derived from this historical survey emphasize: 
first, the geometric representation; second, the algebraic represen- 
tation; and third, the obtaining of the law or rule by an analysis 
of these two representations. Furthermore, in the historical de- 



Suggestions for Presentation of Calculus 49 

velopment of such topics as the derivative it will be seen that there 
are overlapping phases of intuition, demonstration, and application. 
It will be found that frequently there were sound deductions, but 
very often the rigorous proof was not given until many years later — 
and so it may properly be in presenting the subject to pupils at the 
present time. 

THE MATERIAL 

The Use of Graphs. The simplest vehicle for the presentation of 
the theory to beginners is that of graphs. This simplicity is increased 
in the method suggested in this study by the use of the same specific 
curve in the explanations of nearly all the topics of the theory. In 
this way the pupils will have more mental energy left free for con- 
centration upon the new ideas which the calculus presents. These 
graphs are already familiar to the pupils, they are easily understood, 
and they are adaptable, economical, and free from the time-con- 
suming, irritating explanations necessitated by technical material 
taken from the applied fields. A certain amount of material from 
these latter fields will, however, be included in the exercises, par- 
ticularly such as will be easily understood by the pupils. To use 
throughout the explanations of the theory one particular kind of 
material, such as graphs, that is readily understood by the learner, 
not only adds to the simplicity of the treatment but also aids the 
pupils in perceiving the unity that exists throughout the 
theory. 

The material of the course, although given as a connected whole, 
has three divisions. The theory is treated in such a way that it may 
be separated throughout from the exercises and be made a part of 
the course by itself. The first part of the exercises under each topic 
is not too difficult and is selected in such a way as to illustrate 
the ideas that have just been explained and to give the pupils prac- 
tice in the processes involved. The second part of the exercises 
under each topic includes carefully selected interesting applied prob- 
lems. It is left to the option of the teacher to decide which of these 
latter problems the pupils should attempt. 

In the topics that follow in this section the theory of a course in 
the elementary calculus in the high school will be given in fairly 
complete form, but only an illustrative example or two will be given 
of each of the two kinds of exercises just described. 



50 Elementary Calculus in Senior High-School Mathematics 

Second Part 
the method of approach to the subject 

The method of approach to the elementary calculus and to each 
of the topics discussed is left to the judgment of the teacher. A 
teacher who has had training in physics would probably find it 
desirable to introduce the subject of the calculus through some such 
problem as that of a falling body. Another teacher who has not had 
much training in physics would probably find that a talk upon the 
uses of the calculus would be sufficient to arouse immediate inter- 
est in the subject. Many illustrations of the uses of the calculus 
may be found in textbooks on the subject and in technical journals. 
Very likely a single illustration, such as the tinsmith's problem 
(given later under the heading, A Maximum Illustrated by a 
Practical Problem), will give the class sufficient incentive to start 
on its study for obtaining the rule for finding the derivative. Be- 
ginning with the topic, The Function Idea, all the topics are de- 
veloped as a part of the process for obtaining the fundamental rule 
for rinding the derivative of a function. It should be constantly 
kept in mind that the deriving of this fundamental rule is the main 
objective of this part of the theory. 

In the synopsis of the course in the elementary calculus in the 
senior high-school mathematics, the theory is based upon algebraic 
polynomials with the explanations given by means of graphical 
representations. Throughout these suggestions, illustrative exam- 
ples are given which show how the work may be closely allied to 
physics and mechanics. The author believes that this is the prefer- 
able method since it is the shorter road into the mathematical insight 
of the nature of the calculus, since mathematical teachers in general 
(unless they have had a strong preparation in physics) will find this 
the more convenient method, and since also the pupils (unless they 
too have had a good preparation in physics) will be able to grasp this 
method more readily. Furthermore, it is very probable that the most 
difficult thing in the making of a high-school course in the calculus 
is the simplification of the mathematics of the calculus in such a way 
that the deductions made and the statements given will be sound 
and in agreement with results as obtained by the rigorous methods 
of the advanced work in pure mathematics. The author has been 
exceptionally careful to keep this difficulty in mind and is confident 



Suggestions for Presentation of Calculus 5 1 

that the method he has herewith outlined will form a reliable stand- 
ard for the teacher of secondary mathematics to follow if he wishes to 
keep the mathematics of the course in elementary 'calculus correct. 

What is the Calculus? It is likely that the student has heard 
mention made of the calculus in his previous courses in mathema- 
tics. No attempt will be made here to give a complete idea of this 
branch of mathematics. Only an illustration will be given to indi- 
cate in general the nature of the subject. 

Suppose that we had a steel bar n inches in length at a given 
temperature. If we increase the temperature, we will increase the 
length of the bar slightly. We say that the length of the bar has 
taken on an increment. If we decrease the temperature, the length 
of the bar will be decreased slightly. This is also an increment but 
in the negative direction. Hence, we say that the bar varies in 
length as the temperature varies. In this respect the length depends 
upon the temperature; that is, the length of the bar is a function 
of the temperature. 

There are many such related changes in nature. However, these 
related changes need not be according to natural laws only, but may 
be due to other causes. The calculus is based upon a consideration 
of the variations of such function relations. 

A Reason for Studying the Calculus. Thus far the student has 
learned several valuable operations for calculation; i. e., addition, 
subtraction, multiplication, division, involution, evolution, and 
logarithms. The calculus will add two more operations to these, 
namely, differentiation and integration. Without these last two 
operations, much of the mathematics necessary in engineering 
and in the sciences would be impossible. Because of the many 
formulas, the actual calculations in the calculus are not so very diffi- 
cult, but in order to master the mechanics of the subject it is necessary 
that the introductory work should be thoroughly understood. 

The Previous Work in Graphs. Before undertaking the study of 
this proposed course in the elementary calculus, the class is supposed 
to have completed the senior high-school courses in demonstrative 
geometry, algebra, and trigonometry, or their equivalent. If their 
work in graphs did not include the finding of maxima and minima 
values, this topic should be given before the work in this proposed 
course is taken up. 

An illustrative example. A patriotic boy who wished to do 



52 Elementary Calculus in Senior High-School Mathematics 

his part during the world war made up his mind to have a war gar- 
den. There was a vacant lot nearby partly covered with rubbish 
that would give him plenty of space for the garden. A neighbor 
donated a piece of chicken-wire. The boy measured the wire and 
found it to be 48 ft. long. The question was this, how wide and how 
long should the garden be, if rectangular in shape, in order that the 
given wire might enclose the greatest space possible? The boy tried 
different dimensions and used a graph to help him to see the results 
in a better form and to serve as a check on the correctness of his 
work. 

Solution. The width plus the length would be equal to }4 of 48, 
or 24. The boy then made the following table: 



fidth 


Length 


Area 


6 


18 


108 


8 


16 


128 


10 


14 


140 


12 


12 


144 


H 


10 


140 


11 


13 


143 


13 


11 


143 



Then he made the graph from this table. Both his calculation and 
his graph showed him that apparently the dimensions 12 by 12 gave 
the greatest area. 

The method of the elementary calculus for this example is as 
follows. Let x be the width, then the length is 24 — x, and 
x (24 — x) or 24X — x 2 represents the area. The derivative of this 
expression is 24 — 2X. Set the derivative equal to zero and solve 
the resulting expression for x. This gives 12 as the width. Hence 
the dimensions 12 by 12 give the greatest rectangular area that can 
be enclosed by the given wire. 

This latter method in which the derivative is used is an exact 
method and, for more difficult problems, a more convenient method 
for solving problems of maxima and minima. (See tinsmith's prob- 
lem, p. 69.) A number of additional problems of this kind would 
arouse in the class a desire to know how to use the derivative and cre- 
ate a motive for studying the derivative. It should then be pointed 
out that the development of the rule for finding the derivative is the 
first important matter in the beginning work in the proposed course 
in the elementary calculus. After the rule for finding the derivative 



Suggestions for Presentation of Calculus 



53 



has been developed, the number of applications is greatly extended. 
The topics leading up to the rule for finding the derivative of an 
expression are the function idea, independent and dependent varia- 
bles, increment, function notation, and the two steps in the explana- 
tions of the derivative. These will now be discussed in the order as 
here given. 

THE FUNCTION IDEA 

The Previous Training of the Pupils in the Function Idea. Al- 
though the definition of a function may not have been given in the 
preceding mathematics or a symbol given for its representation, 
nevertheless the meaning of a function has been developed to a con- 
siderable degree. 6 The idea of functional relationship is contained 
in much of the preceding mathematics, and the pupils are prepared 
for further instruction in the function idea even if it had not been 
explicitly commented upon heretofore. 

Illustration of the Function Idea. The pupils plot the specific curve 
y = x 2 — 4.x + 5- This recalls to the pupils the ideas they have 
learned about graphs, and brings 
them up to a point where their 
minds are prepared to receive the 
new ideas. 

Then, guided by the teacher, 
they select some suitable point P 
on the curve and draw the ordi- 
nate PM. It is seen that the value 
of the expression y = x 2 — 4X + 5 
at the point P is the length of the 
ordinate PM. This is an impor- 
tant relation that is used later in 
the explanation of the derivative. 



The expression y 



-2 _ 



4* + 5 



may be read "y is an expression 
containing x." However, in the 
language of the calculus, instead of 




Scale:lunft= 2 spaces 

Fig. 10 



6 Secondary School Circular, United States Bureau of Education, No. 5, February, 
1920. The ideas on the function concept contained in this circular are expanded upon 
and it is shown in detail how they may be applied in the classroom in the treatment of 
specific topics, in a report by The National Committee on Mathematical Requirements, 
February, 192 1. 



54 Elementary Calculus in Senior High-School Mathematics 

saying an expression containing x, it is read "y is a function of x" 
The symbol for a function of x is / (x) and is read "function of x." 
Then in the language of the calculus y — x 2 — 4.x + 5 would be 
written f (x) = x 2 — 4.x + 5. This is read "a function of x is 
x 2 — 4X -f- 5." Hence here y and/ (x) stand for the same thing, that 
is, y is identical with/ (x) . Since 3; represents the length of the ordinate 
at some point P, then the length of the ordinate may also be repre- 
sented by / (x) . 

The reason that y in the expression y = x 2 — 4.x + 5 is called a 
function of x is because of the relationship it bears to x. In plotting 
the curve, the values of y were obtained by giving x different values. 
If a small change is made in x in this expression, then there will be a 
corresponding change in the value of y. There are many illustra- 
tions of a relationship like this between two quantities. 

A simple illustration is the relation between the area of a circle 
and the radius of the circle. The area depends upon the value of the 
radius, that is, the area is a function of the radius. In symbols this 
may be represented thus: A = / (R). The pupils should give many 
other illustrations of this kind. 

A concrete illustration. Suppose that we wish to make a rectangu- 
lar garden. The vegetables we wish to grow are to be planted at a 
certain number per square yard. Suppose that the length of the 
garden is fixed by the length of the lot but that the width of the 
garden will depend upon our wishes. Represent the length in yards 
by L and the width by x. Then the number of square yards in the 
garden may be represented by Lx. Since L or the length is fixed, we 
can change the area only by changing the width. A small change in 
the width will produce a corresponding small change in the area. 
Hence the area depends for its value upon the width, and we say 
that the area is a function of the width. In symbols this may be 
represented thus: A =/(#)• 

Examples. Since the number of rings in a cross-section of a tree 
depends upon the age of the tree, we may say that this number of 
rings is a function of the age of the tree. If R represents the number 
of rings and A the age of the tree, then R = f (A). 

The height of the mercury column in a thermometer varies 
as the temperature varies. Hence the height is a function of the 
temperature. We may represent it in this way, H = f (T). 

The quantity of current generated by a dynamo is a function of 



Suggestions for Presentation of Calculus 55 

the rate at which the lines of magnetic flux are being cut. If Q 
represents the quantity and M the number of lines of magnetic flux 
cut per second, then Q = f (M). 

INDEPENDENT AND DEPENDENT VARIABLES. DEFINITION OF A 

FUNCTION 

In the preceding topic it has been pointed out that if in the ex- 
pression y = x 2 — 43; + 5, we give a small increase to x then there will 
be a corresponding change in the value of y. Here y depends for its 
value upon the value that is given x, hence y is called the dependent 
variable. Since x may be given values at will, it is called the inde- 
pendent variable. Everything is now in readiness for the following 
definition of a function. A quantity called the dependent variable 
is a function of another quantity called the independent variable, 
if for any slight change made in the independent variable there is a 
corresponding change in the dependent variable. 

Example. Suppose that a man walks n miles an hour. Then the 
distance, d, that he walks in t hours may be represented by nt. 
Here the distance is a function of the time. This may be represented 
thus: d =/(/). Here t is the independent variable and d is the 
dependent variable. Point out the independent and dependent 
variables in the examples under the preceding sub-topic, a concrete 
illustration. Many additional simple illustrations should be given 
by the pupils. 

INCREMENT 

If in the expression y = x 2 — 4.x + 5> x is given a slight increase, 
then there will be a corresponding change in the value of y. This 
small increase in a variable is called an increment. The term in- 
crement may also stand for a small decrease in the variable. The 
international symbol for the increment of x is Ax. 7 Then the in- 
crement of the value of the expression would be represented by Ay 
since here y stands for the value of the expression. 

In the graph of y = x 2 — 4.x + 5, some point P was selected. 
Suppose that the abscissa of the point P is x and that its ordinate 
is y. Then the coordinates of the point P are x and y. This may be 
represented in this way, P = (x, y). Now suppose that x is given a 

7 Since the differential triangle results from an increment in x, this symbol for the 
increment may be associated with the triangle in that way. 



56 Elementary Calculus in Senior High-School Mathematics 

small increase in value which is represented by Ax. Then there will 
be a corresponding change in the value of y which may be repre- 
sented by Ay. The sum of x plus its increment is represented as 
x + Ax, and the sum of y plus its increment is represented as y + 
Ay. These sums are respectively the abscissa and the ordinate of 
some other point which may be designated as P' . (See the figure 
under the following topic.) Then the coordinates of P' are (x + Ax, 
y + Ay). This relation should be noticed carefully because it is 
used in the explanation of the derivative. 

Example. Suppose that a man walks n miles an hour. Then the 
distance, d, that he walks in t hours may be represented by nt. 
Here the distance is a function of the time and may be represented 
as d = f (t), or it may be represented fully as d = nt. How should 
this last expression be altered to show that the time has been in- 
creased slightly? Then / would have added to it an increment At 
and d would have added to it an increment Ad, and the expression 
may be written as d + Ad = n (t + At). 

Suppose that a railroad company charges a cents per mile traveled. 
Then the cost of a fare depends upon the distance traveled. Let 
d represent the distance and c the cost, then c = f (d). Now if d is 
given an increment, how may this function relation be expressed? 
It may be expressed either as c + Ac = f (d + Ad), or as c + Ac 
= a(d + Ad). 

It has been proved by astronomers by means of very accurate 
observations with powerful instruments that the two pointers of 
the north star are moving very slowly in opposite directions. 8 If 
D represents the distance between them at this instant and T rep- 
resents an instant of time several thousand years hence, write the 
expression representing the distance between these two stars at the 
latter instant. It may be represented as D + AD, since the increase 
in distance is so very slight. The pupils should give additional simple 
illustrations of this kind. 

FUNCTION NOTATION 

Numerical Illustration. In the graph of f (x) = x 2 — 4.x + 5, 
if x is given a particular value, then the corresponding value of/ (x) 
is the length of the ordinate of the point whose abscissa is this par- 

8 Seymour Eaton, How to Locate the Stars, New York, 1891. 



Suggestions for Presentation of Calculus 



57 



ticular value of x. Suppose that the abscissa of a certain point is 
2.5. Then wherever x occurs in the expression, substitute the partic- 
ular value 2.5 that has been given to x. The right-hand part of the 
expression becomes (2.5) 2 — 4 (2.5) + 5. This is a function of x 
when x has been given the particular value 2.5. It is represented in 
this way, 

/(2. 5 ) = (2.5) 2 - 4 (2.5) + 5- 

Then/ (2.5) = 1.25, the result obtained by simplifying the expres- 
sion. Then 1.25 is the ordinate of the point whose abscissa is 2.5, 
since/ (2.5) represents the length of this ordinate. 

Example. Suppose that / (x) = x 2 — 4X + 5. What is the 
ordinate of a point whose abscissa is 3? Wherever x is found in the 
expression substitute the particular value 3. Then / (3) = 3 2 — 
4.3 + 5. Hence the ordinate/ (3) is 2, the result obtained by sim- 
plifying the expression. Additional illustrations of this kind should 
be given by the pupils. 

Increment Illustration. It has already been shown that if some 
point P on the curve represented by y or / (x) = x 2 — 4X + 5 has 
the coordinates (x, y), then / (x) 
represents the value of the ordi- 
nate PM, as shown in Fig. 11. 
Suppose that the abscissa x is 
given an increment Ax, then this 
new abscissa is x + Ax. Hence 
wherever x is found in the expres- 



sion / (x) 



»2 



4x + 5, substi- 




tute this new value x + Ax. Then 
the expression becomes/ (x + Ax) 
= (x + Ax) 2 - 4 (x + Ax) + 5, 
and/ (x + Ax) is the new ordinate. 
If this new[point is designated by P', 
then its coordinates are (x + Ax, 
y + Ay) . This is an important rela- 
tion because of the way it is used in 
the explanation of the derivative. 

Examples. Suppose that a railroad company charges n cents 
per mile traveled. Then the cost of a fare depends upon the dis- 
tance traveled. Represent the cost by c and the distance by d. 



Scale:lunit=2$pace3 

Fig. 11 



58 Elementary Calculus in Senior High-School Mathematics 

Then c = / (d), that is, the cost is a function of the distance. Now 
if the distance is increased slightly, how should the expression 
c = f (d) be altered? It should be written c + Ac = f (d + Ad). 
The pupils should give additional illustrations of this kind. 

Suppose that f (x) = x 2 + x + I . What is / (z) ? What is 
f(a + b)? f(z)=z 2 + z+i. f( a + b) = (a + b) 2 + (a + b) 
+ I. 

Other Ways of Representing the Function Besides f (x) . Suppose 
that in the same problem, there are two different expressions in- 
volving x. One of these may be represented by / (x) and the other 
by g (x) or some other letter before the parentheses. To distinguish 
these expressions in talking about them they are read "the /function 
of x" and "the g function of x." 

Example. In solving a practical pioblem these two different 

functions of x were involved, 

100,000 , , ne (rx 2 + nR) — 2rx- nex 

-r-55* and 



3 (x + i) (rx 2 + nR) 2 

How may these two functions be indicated briefly and yet be 
distinguishable? This may be done by representing the first one 
by/ (x) and the second by h (x). 

Frequently a simple expression is represented by some other 
form than / (x) . 

Example. Suppose that g (x) = x 3 — I. What is g (3)? g (3) = 
3 3 — 1 = 26. It should be noted that if a function as x 3 — 1 is 
represented as the g function of x, then the g before (x) in the 
symbol g (x) must not be changed. Thus, if g (x) = x z — 1, then 
k (x), or the k function of x, is a function different from x z — 1. 

Suppose that s (t) = t 2 — 4/ + 5. Find s (o), s (1), s (2), 5 (3), 
and 5 (4). These values are 5, 2, 1, 2, 5, respectively. 

Suppose that/ (y) = y 2 — c 2 . Find/ (i-c). Ans. 1 — 2c. 

Suppose that/ (B) =4,8 + 3. Find/ (A). Ans. ^A +3. 

Suppose that/ (z) = sin z — cos z. Findf (x). Ans. sin x — cos x. 

Suppose that/ (u) = tan u. Find/ (0.5236). Ans. tan 0.5236. 

Suppose that/(#) = # 3 + 1. Find/ (5). Ans. 126. 

Suppose that / (w) = w b + w A + w z + w 2 + 2W + 17. ^4^5. 23. 

Suppose that/ (x) = x 4 — 5X 2 . Find/ (3). ^4ws. 26. 

The pupils should give many additional examples of this kind 
in which the function is represented by other letters preceding the 
parentheses in the symbol, as m (x), F (z), and so on. 



Suggestions for Presentation of Calculus 



59 



THE DERIVATIVE 

The derivative is the most important topic in this part of the 
elementary calculus. The explanation of the derivative as given 
here is based upon the ideas that were developed in the preceding 
topics, and for this reason those topics were developed slowly and 
carefully. 

In the explanation of the derivative, the language throughout has 
been very carefully chosen in order that the mathematical course 
may stand out clearly. The figure used in the explanation should 
have on it the fewest possible lines and these should show up all the 
necessary parts very distinctly. The first step in the meaning of 
the derivative will now be given. 

The First Step. The attention of the class is called to two pre- 
liminary facts that will later be used in the explanation of the figure. 





Fig. 13 



One of these is that if through any point P on a straight line AB, 
another line CD is drawn, then CD may be moved about the fixed 
point P until it coincides with the line AB. The second fact is that 
in a right triangle EKF the slope of the line EF is the tangent of the 
angle at E. Also, in the explanation it is well to use the word slope 
instead of tangent in order to avoid confusing it with the tangent 
line. 

The pupils now plot/ (x) = x 2 — \x + 5. Guided by the teacher, 
they select a suitable point P and through it draw the line PT 



6o Elementary Calculus in Senior High- School Mathematics 



tangent to the curve. Next they select a convenient point P', 
and through P and P' they draw the secant PP'S. Then they 
draw the ordinates PM and P'M', and also PL parallel to the 
X-axis. 

Now consider the right triangle PLQ. It shows that the slope of 
the tangent line PT is the ratio QL/PL. Now consider the right 

triangle PLP' '. It shows that the 
slope of the secant PP'S is the 
Uyfa iafc/3 1 1 / / ratio P'L /PL. 

-X -=r>f~ — Now imagine that the point P' 

moves along the curve toward P, 
then the secant PP'S will more 
nearly take the position of the 
tangent line PT. As the point P' 
gets nearer and nearer to the point 
P, the position of the secant line 
will approach yet more nearly to 
the position of the tangent line, 
and the slope of the secant line 
will approach yet more nearly to 
the slope of the tangent line. In 
fact, under these conditions the 
position of the tangent line is 
the limit of the position of the 
secant turning about the point P, and the slope of the tangent line 
is the limit of the slope of the secant. 

Next, turn this into the notation of the calculus. Represent the 
coordinates of P by (x,y). Then by giving x an increment Ax, the 
coordinates of P' will be (x + Ax, y + Ay). This idea was developed 
under the preceding topics. Then, since PL = Ax and P'L = Ay, 
the slope of the secant is Ay /Ax. Also, as the point P' moves along 
the curve toward P, Ax approaches zero as its limit. The symbol 
for expressing the limit of the slope of the secant as Ax approaches 

zero as its limit is nm — • This symbol is the geometric ratio 
Ax->o Ax 

form of the derivative. This is read "the limit of the ratio Ay to Ax 

as Ax approaches zero as its limit." 

The Symbol f (x). At this early stage in the learner's knowledge 

of the calculus, it would be well to use only /' (x) to represent the 




Sca/e: lunit* 2 spaces 

Fig. 14 



Suggestions for Presentation of Calculus 



61 



derivative of the function of x. 9 This form is used to a large extent 
in the more advanced calculus and hence it is desirable from the 
point of view of increased knowledge of mathematical language. 

Then the derivative may be expressed as /' (x) = lim — • 

x ->- o Ax 

This symbol is a quick way of denning the derivative, gives a 
desirable point of view, and is not an entirely new way of present- 
ing a definition since the definitions of the trigonometric functions 
were also explained by stating that they mean such and such things 
merely because it is agreed that they shall mean those things. 
The Second Step. This second step in the meaning of the deriv- 
ative is the development of Cauchy's fractional form of the de- 
rivative. The same curve is used 
as in the preceding step. The or- 
dinate of P, the point of tangency, 
PM"equals/(x). The ordinate of the 
moving point P' equals/ (x + Ax). 
Now / (x) represents the value of 
the original function and/ (x + Ax) 
represents the value of the func- 
tion after x has been given an 
increase. Then the difference be- 
tween these two ordi nates is the 
increment of the entire function. 
This difference is/ (x + Ax) — / (x) 
and represents the increment of 
the entire function also. Hence 
Ay =f (x + Ax) — / (x) , since under 
the sub-topic, The First Step, it 
was shown that Ay also represents the increment of the entire 
function. Substitute this new form of Ay in the definition of the 

derivative as represented bv um — 

Ax ->- o Ax 




\te;±A*) 



Scale. :lunit=2 spaces 

Fig. is 



9 Throughout the synopsis, the notation has been selected that will most nearly 

dy 
fulfill the requirements of the principle of no exception. The use of the form — 

ax 

here would be a flagrant transgression of this principle, but after the explanation of the 

dy 
differential, the form — is then permissible since it may then be regarded as a fraction. 
CLx 



62 Elementary Calculus in Senior High-School Mathematics 

This gives Cauchy's fractional form of the derivative, namely, 

/' («) = lim /(* + A*)-/W 
Ax ->- o Ax 

77ze Fundamental Rule. The steps of the fundamental rule for 
finding the derivative are obtained directly from an analysis of 
Cauchy's fractional form of the derivative. / (x + Ax) stands for the 
function after x has been given an increment. / (x) is the original 
function. The difference between these two gives the increment 
that the entire function has assumed. This difference is divided 
by the increment of the independent variable. The limit of this 
resulting fraction is the derivative of the function. 

Hence the fundamental rule for finding the derivative is: 

a. Find the expression after x has been given an increment. 

b. Subtract the original function from this expression. 

c. Divide the resulting difference by the increment of the inde- 
pendent variable. 

d. Take the limit of this quotient as the increment of the inde- 
pendent variable approaches zero as its limit. 

The limit of this quotient is the desired derivative. 
Application of the Fundamental Rule. Suppose that / (x) = $x + 2. 
Find the derivative of / (x) . 

Give x an increment. The expression then becomes 

a. 3 (x + Ax) + 2. This simplified gives 

3x + 3 -Ax + 2. From this subtract the original 

function 3X + 2. This gives 

b. 3 -Ax Divide this expression by Ax. This gives 

c. 3-Ax / Ax = 3. 

d. Take the limit of this quotient as Ax approaches zero 
as its limit. Since 3 is a constant, it is not affected by 
changes in x. Hence the limit is 3. Therefore the deriv- 
ative of 3X + 2 with respect to x is 3. This may be represented 
by f(x) =3. 

Examples. By means of the fundamental rule, find the deriv- 
ative of the following: 

/ (r) = r 2 + 3. Ans.f (r) = 2r. 

f ( z ) = 3S 2 + 2# A ns ^ f ( 2 ) = 6z + I. 

/ (y) = y* + I. Ans.f (y) = 3;y 2 . 



Suggestions for Presentation of Calculus 63 

The pupils should solve additional simple examples. Addi- 
tional applications of the fundamental rule for finding the deriv- 
ative will be made in the problems on the slope of the tangent line 
and in the derivation of formulas. 

Remarks. It is hoped that the unity, simplicity, and definite- 
ness in the development of the course thus far is evident. The 
pupils realize that the ideas of function, increment, independent 
and dependent variables, function notation, and derivative are 
not unrelated concepts. The pupils are not lost in the mathemat- 
ical thought being developed by a discussion on side-topics of theory 
that are beyond their comprehension, and they are not tired out 
mentally by a flood of detail given merely to make the discussion 
exhaustive. Especially is this improvement evident in the devel- 
opment of the idea of the derivative by the use of the three distinct 
steps: the derivative expressed as the limit of a geometric ratio, 
Av/Ax; its interpretation into Cauchy's fractional form of the 
derivative; and the analysis of this fractional form into the four 
steps of the fundamental rule for finding the derivative. 

It is desirable that the pupils should have a sufficient amount of 
exercise in finding the derivative of functions by means uf the funda- 
mental rule in order that the meaning of the derivative may be 
impressed upon their minds. After they once begin to use formulas 
they make very little use of the fundamental rule, but this rule 
is necessary in order to derive the formulas. Problems on the slope 
of the tangent form suitable material for exercises in the application 
of the fundamental rule. Other suitable problems are found in any 
standard college textbook on the calculus. 10 

THE SLOPE OF THE TANGENT LINE 

In the discussion of the first step under the derivative it was shown 
that the slope of the tangent line is the limit of the slope of the 
secant, and also that the limit of the slope of the secant is one way 
of defining the derivative. Hence the value of the derivative of a 
function at a given point on the curve is the slope of the tangent 
line at that point. 

10 W. A. Granville, Differential and Integral Calculus, p. 31, Boston, 1911. 



64 Elementary Calculus in Senior High-School Mathematics 



Plot the graph of / (x) = x 2 — 4.x + 5. Find the slope of the 
tangent line at the point P = (2.5, 1.25). To do this, first find the 

derivative of/ (x) by means of the 
fundamental rule. This gives 
fix) = 2x — 4. The value of this 
derivative at the point P is the 
slope of the tangent line at this 
point. The value of the derivative 
at the point is found by substitu- 
ting into it the coordinates of the 
point. Thus, \ix — 4] x y Z i.l s = 
2 (2.5) -4=1. 

The value of y does not need to be 
considered here since y does not 
occur in the derivative. Hence the 
slope of the tangent line at the 
point P = (2.5, 1.25) is 1. Since 
the tan 45 is 1, the angle which 
the tangent line makes with X- 
axis is 45 . 




Scale: lunit=2 spaces 



Fig. 16 



Example. Plot the graph of / (x) 



.2 _ 



4x + 5. Find f(x), 



the slope of the tangent line at the point whose abscissa is x = 4, 
and also approximately the number of degrees made by the tangent 
line and the X-axis. Plot the curve. 

Ans. (a) 2x — 4; (b) its slope is the ratio 4:1; (c) the angle is 
75° 57-8'. 

Suppose that/ (2) = - z 2 - 43 + 5. Find (a) /'(z); (b) the 
slope of the tangent line at the point whose abscissa is x = 
— 4; (c) approximately the number of degrees in the angle. Plot 
the curve. 

Ans. (a) — 2z — 4; (b) its slope is the ratio 4:1; (c) the angle is 
75° 57-8'. 



Suppose that / (s) 



:2 _ 



45 + 3. Find (a) /' (s); (b) the s^pe 



of the tangent at the point whose abscissa is x = 4; (c) approxi- 
mately the number of degrees in the angle. Plot the curve. 

Ans. (a) 2s — 4; (b) its slope is the ratio 4:1; (c) the angle is 
75° 57-8'. 

The pupils should work a sufficient number of similar ex- 
amples. 



Suggestions for Presentation of Calculus 65 

FORMULAS 

The Differentiation of the Function ku with Respect to x. The 
process of finding the derivative of a function is called differentia- 
tion. The formulas for this purpose are found by means of the appli- 
cation of the fundamental rule. This method will now be illus- 
trated by finding the derivative of ku with respect to x. 

Give the variable u an increment. This gives 

(1) k (u + Aw). This simplified gives 

ku + k-Au. From this subtract the original function 
ku. This gives 

(2) k-Au. Divide this expression by Ax. This gives 

(3) k-Au/Ax. Take the limit of this quotient as Ax approaches 

zero as its limit. This gives 

(a) v h A^ 

vhv urn r — . Since k is a constant, it is not affected by 
Ax-X) Ax 

changes in x. Hence k may be written before the symbol for the 

A 7/ 

limit. Then the expression becomes k- lim — . The latter part of this 

Ax->-o Ax 

expression is the derivative of u with respect to x. Hence the 

derivative of a constant times a variable is equal to the constant 

times the derivative of the variable. 

Examples. If / (/) = kt, find the derivative of kt with respect 
to x by means of the fundamental rule. 

The pupils should derive as many additional formulas as the 
teacher may decide. It is, however, more important that the pupils 
should learn how to use the formulas. 

Examples. Make use of the formulas in solving the following. 11 

Find the derivative of 5X 3 ; 4J 2 + 3/ + 1 ; av 2 + bv + c; jz* + 4-z 3 
+ z; 4g + 3; 4X 2 + 3X + 2; 2s 3 + 2s 2 + 3; aw 2 + bw + 2; u 11 ; 
ht a + 3. 

Applications of the Derivative. There is a wide use made of the 
derivative in practical problems, and illustrations of these may 
be found in almost any standard college textbook on the calculus. 12 
The use of the derivative for finding the slope of the tangent line 
to a curve has already been shown in this course. Its use will now 
be shown in the subject of maxima and minima. 

11 Use list of formulas as found in any standard college textbook on the calculus. 

12 William F. Osgood, Elementary Calculus, Chap. Ill, New York, 1921. 



66 Elementary Calculus in Senior High-School Mathematics 



MAXIMA AND MINIMA 

Maxima and minima form not only a suitable subject for appli- 
cations of the derivative but also a valuable one from the stand- 
point of later work in mechanics and other applied subjects. How- 
ever, the explanations of maxima and minima here will be given 
by means of graphs in keeping with the idea that the subject- 
matter of the theory of this course should be dosely related and of 
the same kind throughout. 

A Minimum- in a Graph. Plot the graph of/ (x) = x 2 — 4.x + 5. 
Its lowest point is found by inspection. It is the point P = (2, 1). 

Draw a tangent to the curve at 
this point. It is seen that this 
tangent line is parallel to the 
X-axis. Hence the slope of the 
tangent line at a point that gives 
a minimum value of the function 
is zero. Also, the value of the 
derivative of the function at this 
point is zero, since the derivative 
at a point equals the slope of 
the tangent line at that point. 
Now reverse this process. Find 
the derivative of the function 
/ (x) = x 2 — 4.x + 5. The deriv- 
ative /' (x ) is 2 x — 4. Set this 
derivative equal to zero and solve 
for x. 2x — 4 = o. x = 2. This 
is the abscissa of the point P. 
Find the ordinate corresponding 




Scale: 1 unit =2 spaces 



Fig. 17 



to this abscissa. Since / (x) 



4X + 5, then / (2) = 2 2 — 4-2 



+ 5 = 1. This is the ordinate of the point P. Apparently this is 
a method that will give the coordinates of the point where the func- 
tion has a minimum value. It will be shown in the next topic 
that this same method is used for finding the point where the 
function has apparently a maximum value. Hence it is necessary 
to find the way to tell whether the function is a minimum or a 
maximum at the point whose abscissa is the value of x found by 
this method. The way to tell the difference will now be shown. 



Suggestions for Presentation of Calculus 67 

In the graph of / (x) = x 2 — 4.x + 5, the point where the func- 
tion is a minimum is P = (2, 1). This means that the ordinate 
at the point P is less in value than any other ordinate very close to 
it. Both positive and negative numbers must be considered. 
Now use this check on the ordinate of P and see if it really is a 
minimum value of the function. At P, x = 2. This is called a 
" sign-post " value of x since it shows the way to a minimum or a 
maximum value of the function. / (2) = 1. Suppose that x = 1.9, 
then.,/ (1.9) = (1.9) 2 - 4 (1.9) +5 = 3-6i - 7-6 + 5 = i.oi, a 
value a little greater than 1. It will be found that/ (2.1) will also 
give a value a little greater than 1. Hence 1 or the ordinate of P 
is a minimum value of the function. 

A Minimum Illustrated by a Practical Problem. In practice the 
nature of the problem will often indicate whether the sign-post 
value of x leads to a minimum or a maximum value of the function, 
as is shown in the following printer's problem. 

Since the cost of a certain kind of job in printing varied from 

$100 to $200, depending upon the number of electrotypes used, 

the printer wished to know the most economical number. The 

conditions of the problem are as follows: 200,000 (P) prints are 

required ; 1200 (S) prints per hour is the speed of the press; $2.00 

(R) per hour is the cost of running the press; 55 cents (E) each is 

the cost of the extra electrotypes needed after the type is once set 

up. Required the number of electrotypes (x) that should be used 

to secure the minimum cost (C). The problem is evidently one in 

determining the minimum value of C by the use of the differential 

calculus. Here C is a / (x) . Solution: 

„ , N 200,000-200 . 100,000 , . . 

/(*) = ; — ; — : + 55* = — — ; — - + 55*> where x is the 

1200 (1 + x) 3 (x + 1) 

number of electrotypes. By differentiation, /' (x) = — 100,000/3 

times + 55 = o. This derivative is set equal to zero 

ix + i) 2 

in order to find the sign-post value of x. Hence x = . \/66 — I = 

3 
23.6 +. This value of x will make the function a minimum. There 
is no maximum value of the function since it is seen that as x 
becomes larger and larger the value of the function becomes larger 
and larger. 



68 Elementary Calculus in Senior High-School Mathematics 



Therefore the most economical number of electrotypes to use 
for this purpose is 24. 13 

Examples. A Labrador missionary-doctor is in his rowboat 4 
miles from shore. A messenger informs him that a man 6 miles down 
the coast is in extreme need of his services. The doctor can row at the 
rate of 3 miles per hour. On shore, the doctor can make an average 
speed of 4 miles an hour on his skiis. How far down the shore should 
he land in order that he may reach bis patient in a minimum 
length of time? The distance will be found to be 4^ miles. 

An architect wishes to design a two-story type-cottage with 
pitched roof such that the cross-section of the room space on the 
second floor will be 12 ft. wide and 6 ft. high. What will be the mini- 
mum length of the rafters? Ans. 16.968 ft. or 17 ft. Suggestion. Con- 
sider the right triangle formed 
by a rafter, the perpendicular 



faHlxts distance from the ridge to the 
second floor, and half the width 
of the cottage. The cross-section 
of half the room space will be 
a square inscribed in this right 
triangle with a side 6 ft. in length. 
Then use the proportions result- 
ing from the properties of similar 
triangles. 

A Maximum in a Graph. Plot 
the graph of / (x) = — x 2 — qx 
+ 5. Its highest point is found 
by inspection of the figure. It 
is the point P = (— 2, 9). Draw 
a tangent to the curve at this 
point. It is seen that this tan- 
gent line is parallel to the X- 
axis, and hence the slope of the 
tangent line at a point that gives 
a maximum value of the function 
is zero. Also, the value of the 
derivative of the function at this point is zero, since the derivative 

13 Edgar E. DeCou, "A Practical Printer's Problem", American Mathematical Monthly, 
V. 27, Nov. 11, 1920. 



r 



Scale: 1 unit=2 spaces 



Fig. 18 



Suggestions for Presentation of Calculus 



6 9 



at a point equals the slope of the tangent at that point. Now re- 
verse the process. Find the derivative of the function/ (x) = — x 2 

— 4X + 5. The derivative/' (x) = — 2x — 4. Set this derivative 
equal to zero and solve for x. 

— 2x — 4=0. x = — 2. This is the abscissa of the point P. 
Find the ordinate corresponding to this abscissa. Since/ (x) = — x 2 

- 4X + 5, then / (- 2) = - (2) 2 - 4 ( - 2) + 5 = 9. This is 
the ordinate of the point P as found by inspection. Apparently this 
is a method that will give the coordinates of the point where the 
function has a maximum value. It has, however, already been shown 
that this same method is used for finding the point where the func- 
tion has a minimum value. It was also shown how to use the sign-post 
value of x to tell if it will give a minimum value of the function. How 
to tell if it will give a maximum value of the function will now be shown. 

In the graph of / (x) = — x 2 — 4.x + 5, the point where the 
function is a maximum is P = (— 2, 9). This means that the ordi- 
nate at the point P is greater in value than any other ordinate very 
close to it. Both positive and negative numbers must be considered. 
Now use this check on the ordinate of P and see if it really is a 
maximum value of the function. At P, x = —2, and the ordinate 
/(— 2) is 9. Find the ordinates /(— 2.1) and /(— 1.9). It is 
seen that each of these is less than the ordinate/ (—2). Hence P is 
a point where the function has a maximum value. 

A Maximum Illustrated by a Practical Problem. Suppose that a 
tinsmith has on hand ends of strips of tin left over from some job. 
He wishes to make a box with each piece for holding such things as 
nails of different sizes. Each piece is 15 inches by 8 inches. The 
boxes shall be rectangular in shape and shall contain the greatest 
volume possible. They are made by cutting out a square piece from 
each corner and then turning up the sides. What should be the size of 
the piece cut out of each corner? 
First we must form the expres- IS 

sion of which we wish to find the 
maximum. Let x = the size of 
the square to be cut out of each 
corner. Then 15 — 2x = the 
length of one side of the box, and 
8 — 2x = the width of the box, 
and x = the depth of the box. fig. 19 




70 Elementary Calculus in Senior High-School Mathematics 

Then the volume is x (8 — 2x) (15 — 2x). This is the expression 
of which we are to find a value of x so that the volume will be a 
maximum. Expanding this expression, we have/ (x) = 4.x 3 — 46X 2 
+ 120X. Its derivative is/' (x) = I2x 2 — 92* + 120. Set the deriv- 
ative equal to zero^and^solve for x. 

I2# 2 — 92* + 120 = o, 

33c 2 — 2$x + 30 = o. 

x = (23 ±V529 - 36o) /6 = 6, or l%. 

We see that x = 6 inches is not possible. Hence we select x = 1% 
inches. Then if the box is constructed according to these conditions, 
its volume will be the maximum. Check this result by plotting 
/ (x) = x (8 — 2x) (15 — 2x). This factor form makes it easier 
to substitute different values for x for plotting. 

Examples. Given the conditions the same as in the preceding 
problem except that the length is 16 inches and the width is 6 inches. 
Find the value of x that will make the volume a maximum. . It will 
be found that this value of x is one and one-third inches. 

A farmer wishes to build a concrete cylindrical cistern that shall 
contain 5,000 cu. ft. of water. What should its dimensions be in 
order that the cost of the material will be a minimum? The surface 
is made up of the lateral surface and the base of the cylinder. It 
will be found that the radius should be made equal to the height 
or 1 1.7 ft. 

Maxima and Minima in Graphs. A graph may show a number of 
points where the function has a maximum value or where it has a 
minimum value. An example illustrating this will now be 
given. 

Find the points where the function / (x) = x 3 — 2x 2 — 5^ + 6 
has either a maximum value or a minimum value. Its derivative is 
/' (x) = 3X 2 — 4.x — 5. Set this equal to zero and solve for x. 
Then x = 2.1, or —0.8. Here there are two sign-post values of x. 
First consider the value — 0.8. If the abscissa x is — 0.8, then 
the ordinate / ( - 0.8) is (- 0.8) 3 - 2 (- 0.8) 2 - 5 (- 0.8) + 6, 
or 8.2. That this is a maximum value of the function may be seen 
by finding ordinates corresponding to values of the abscissa x that 
are a little greater and then a little less than — 0.8. It will be found 
that these ordinates are less than the ordinate /(— 0.8). Hence 
'/ ( — 0.8) is a maximum value of the function. 



Suggestions for Presentation of Calculus 



71 



If the abscissa is x = 2.1, then the corresponding ordinate/ (2.1) 
is a minimum value of the function. Verify this by the usual 
checking method. Hence/ (x) = 
x s — 2x 2 — 5X + 6 has a maxi- 
mum value / (— 0.8) for the 
value of x = — 0.8, and a mini- 
mum value / (2.1) for the value 
of x = 2.1. 

Suggestions for Plotting the Curve 

The important points in plot- 
ting a graph are the place where 
it crosses the/ (x)-axis, the place 
where it has a maximum or a 
minimum value, and the place 
where it crosses the X-axis. An 
illustration of how to do this will 
now be given. 

Plot the graph of / (x) = x 3 

— 2x 2 — 5X + 6. 
At the place where the curve 

crosses the / (x)-axis, the ab- 
scissa x = o. Since / (o) = o 3 

- 2 (o) 2 - 5 (o) + 6 = 6, this 
is the place where the graph 
crosses the/ (x)-axis. 

The function has a maximum 
value at the point (— 0.8, 8.2) 
and a minimum value at the point (2.1, —4). 

The places where the function crosses the X-axis are the roots of 
the equation. These are x = 1, x = 3, and x = — 2. 

The general shape of the graph may now be drawn, as shown in 
Fig. 20. Additional points may. be found if a more careful graph is 
desired. 

Example. Find the values of x for which the foDowing function 
will be a maximum and for which it will be a minimum. The function 
is / (x) = X s — 6x 2 + 12. It will be found that the function has a 
maximum value when x = o, and a minimum value when x = 4. 




Scale: I unit* 2 spaces 

Fig. 20 



72 Elementary Calculus in Senior High-School Mathematics 



THE DIFFERENTIAL 

Reasons for Studying the Differential By means of the differential 
we pass from differentiation to integration and reverse the process. 
A knowledge of the differential also enables pupils to extend the 
skill they have thus far acquired to other problems. The idea of the 
differential is made simpler by explaining it in such a way that pupils 
are able to visualize it. 

Geometrical Representation. Plot the graph of y = x 2 — ax + 5. 
The slope of the tangent line is QL /PL, and the slope of the secant is 

P'L /PL. These are used here 
under the same conditions under 
which they were used in the expla- 
nation of the derivative. The co- 
ordinates of P are (x, y) and of P' 
are (x + Ax, y + Ay). Then P'L 
represents Ay. It has been shown 
that the limit of the slope of the 
secant is the slope of the tangent 
line, as Ax approaches zero as its 
limit. Now if the point P' is 
taken in some particular position 
on the curve, as shown in the ac- 
companying illustration, then Ay 
is made up of two parts, P'Q and 
QL. Only the latter, QL, will be 
considered here. The increment 
of the entire function is P'L = Ay. 
The part QL of this increment is called the differential of the func- 
tion. It is represented by the symbol, dy, and is read "the differential 
of y." The increment of the independent variable PL = Ax. This 
is equal to dx, as is shown in the more advanced work in the cal- 
culus. However, it is seen from the explanation here that dy does 
not equal Ay. It will be shown in the next sub-topic what dy really 
does equal. Since these quantities, dy and dx, are finite, they may 
be substituted for their equals in the fraction, QL/PL, which rep- 
resents the slope of the tangent line. Hence the slope of the 
tangent line at a particular point is dy /dx. 

What dy Equals. It has previously been shown that the slope of 




Scale: I unit =2 spaces 



Fig. 21 



Suggestions for Presentation of Calculus 73 

the tangent line at a particular point of the curve is the derivative of 
the function or the value of f (x) at that particular point. In the 
preceding sub -topic it was shown that the slope of the tangent line 
is the value of dy jdx at that point of tangency. Hence the value 
of dy may be found by setting these two quantities equal to each 
other and solving the equation for dy. Thus, 

dy/dx=f(x), then dy = f'(x)-dx. 

Hence the differential of a function is found by multiplying the 
derivative of the function by the differential of the independent 
variable. If this method is adhered to, it will not be necessary to 
make use of a separate table of differentials. 14 

Examples. Find the differentials of the following functions. 

f(x) = 5X 2 . Then df(x) = loxdx. 

f(t) = 4^ - 2>t 2 - 1. Then df(t) = \2tHt - 6tdt. 

Find the differentials of the following: 



5* 2 . 






4 / 3 - 


3t 2 ~ 


~ I, 


av 2 — 


b, 




72 s - 


z, 




kt, 






y 2 gt, 






ht a - 


2t - 


- 3, 


cx n ~ 


1 

> 





Ans. 


loxdx. 


Ans. 


I2t 2 dt - 6tdt. 


Ans. 


2avdv. 


Ans. 


2\z L dz — dz. 


Ans. 


kdt. 


Ans. 


y 2g dt. 


Ans. 


ahf ~ l dt - 2dt. 


Ans. 


(n-i)cx n ~ 2 . 



The pupils should solve many additional examples of this 
kind. 

INTEGRATION 

A Reason for Studying Integration. One of the reasons for study- 
ing integration is that it enables one to find the areas of figures which 
cannot be found by the ordinary geometric methods. Plot the curve 
/ (x) = x 2 — \x + 5- Erect ordinates at x = a and x = b. How 
can the area enclosed by these ordinates, the curve, and X-axis be 
found? This will now be shown. 

14 The general treatment of the differential of which the method here given is an 
original adaptation is found in Granville's Calculus, p. 132. 



74 Elementary Calculus in Senior High-School Mathematics 




At the point P the abscissa x equals a. Give x a small increment. 
Represent this by dx. This may be done since under the topic of the 

differential it was pointed out that 
i \f(X)-qxis , th e increment of the independent 

variable x equals dx. At the right- 
hand extremity of dx erect an or- 
dinate. Then this small strip of 
the area is almost rectangular in 
shape with a base equal to dx and 
an altitude approximately equal 
to the ordinate / (x). Then the 
area of this strip is approxiately 
f (x)-dx. Next divide the remain- 
der of the area into strips each 
with its base equal to dx. Let x assume these different points of 
division considered as abscissas along the X-axis. Then for each of 
these values of x the corresponding values of the ordinates will be 
/ (x), and the area of each of these strips will be represented by 
/ {x)-dx. The sum of all such strips is approximately the entire area 
desired. In fact, it is proved in the more advanced calculus that 
the limit of this sum as dx becomes infinitesimally small is the 
exact area of the figure. This limit of the sum of these strips is 
represented by ff (x)dx. The mark f is a modification of the 
letter S, the first letter of the word sum. The symbol is read "the 
integral of / (x)dx." The word integral means the whole and as 
used here it represents the sum of all the parts. In case there are 
definite limits to the value of x as in the illustration, the symbol is 



Scale: 1 unit *5 spaces 

Fig. 22 



written 



/ / (x)dx, where a stands for the lower limit of x and b 

stands for the upper limit of x. 

Integration the Reverse of Differentiation. Suppose that we consider 

onlv the direction along the positive end of the X-axis. Take n 

points of division on the line be- v 

- - • 1 , . , ,, dx -a 
with zero and with the 



ginning 



Fig. 23 



distance between any two succes- ' 2 

sive points equal to dx. Then the 

value of Xn would be the integral of dx, that is, the sum of all these 

elementary divisions. Now let x represent in general any point 

along the X-axis, then the value of x may be represented by fdx. 



Suggestions for Presentation of Calculus 75 

This may be written x = fdx. Apparently the quantity under the 
integral sign is the differential of the expression to which the 
integral is equal. In fact, it is true that the value of the integral is 
the expression which when differentiated will give the quantity 
under the integral sign. Thus, it is seen that integration and dif- 
ferentiation are reverse processes, somewhat like division and multi- 
plication, since one may be obtained from the other. 15 The method 
for finding the integral when the differential is given will now be 
taken up. 

A First Attempt at a Table of Integrals. First put down some 
simple graded functions and their differentials, as 
dx is the differential of x, 
xdx is the differential of }4x 2 , and so on. 

Since the function from which we obtained the differential is the 
integral of that differential, the table may be written, 
x is an integral of dx, 
yix 1 is an integral of xdx, and so on. 
However, if the function from which the differential was obtained 
contains a constant term, this term disappears entirely in the 
differential since the differential of a constant is zero. Hence when 
we reverse the process for finding the integral, there is no way of 
knowing what this constant term was in the original function. 
Thus, 

the differential of x + 5 is dx + o, and 
the differential of x + 7 is dx + o. Then the 
integral of dx is x plus or minus a constant. Hence in reversing 
the process of differentiation, the constant term in the integral 
is indeterminate. Integrals in which the value of the constant 
term is not known are called indefinite integrals. Sometimes the 
conditions of the problem are such that there are no indeterminate 
constant terms in the integral and the integral is taken between 
definite limits of the independent variable. Such integrals are called 
definite integrals and will be explained later. 

If the differentials and their corresponding integrals are arranged 
in order, a general formula may be derived. An illustration of this 
will now be given. (Bernoulli's table.) 

15 This method was suggested by Professor D. E. Smith in discussing an experiment 
on integration in the Horace Mann School, Teachers College. 



76 Elementary Calculus in Senior High- School Mathematics 

The integral of adx is ax ± a constant, 

The integral of axdx is yiax 2 ± a constant, 

The integral of a# 2 dx is ^ax 3 ± a constant, 

The integral of axHx is ^ax 4 ± a constant, and so on. 

Hence the integral of axPdx is .x^ +1 ± a constant. 

p + I 

The pupils should be given practice in the use of the integral 
tables in finding the integrals of given differentials. 

Examples. Find the integrals of the following: 

loxdXy Ans. $x 2 + c. 

$t 2 dt - 2tdt, Ans. t z - t 2 + c. 

6avdv, Ans. $av 2 + c. 

\$z 2 dz — 2dz, Ans. 5s 3 — 2z + c. 

af ~ l dt - 2tdt, Ans. t a - t 2 + c. 



THE DEFINITE INTEGRAL 

In the beginning of the discussion on integration it was shown 

f b 
that / / (x)dx represented the area bounded by the curve, the X- 

axis, and the ordinates erected at a and b on the X-axis. Then the 
range of values that x assumes is the segment of the X-axis from a 
to b, or b — a. To find the value of the integral first substitute b for 
x in the integral and then a, the difference between these two results 
taken in this order is the value of the integral. This too is the value 
of the area which was to be found. The constant of integration does 
not need to be considered in the definite integral since it cancels 
out. 

Example. In the illustration just used, suppose that a = 1, 
b = 3, and/ (x) = x 2 — 4.x + 5. Use Fig. 22. 

Solution. Substitute the given values in the formula for the area 

r b 

which is / / (x)dx. This gives 
/ (x 2 — 4X + 5) = / x 2 dx — / 4xdx + / $dx = - — -x 2 + 5x • 
This last form shows how to indicate the limits that are to be sub- 



Suggestions for Presentation of Calculus 77 

stituted in the result. The rule for substitution is : "Substitute the 
upper limit for the variable in the expression in brackets and then 
subtract from this result the result obtained by substituting the 
lower limit for the variable in the expression in brackets." 

Thus, (-2- - 2- 3 2 + 5-3) - (X - 2 + 5) = (9 - 18 + 15) - 

3 
(X- 2 + 5) = 6 - 3^ = 2%. 

Hence the area is 2^3 square units. Check this result by means of 
squared paper. 

Examples. Suppose that the conditions are the same as in the pre- 
ceding problem except that the limits are from the ordinate 2 to 
to the ordinate 4. Find the area bounded by the curve, the X-axis, 
and these two ordinates. 

Suppose that f (x) = — x 2 — 4 x + 5. Find the area bounded 
by this curve and the X-axis between the limits x = — 5 and 
x = 1. 

Suppose that/ (x) = x z — 2x 2 — 5x + 6. Find the area bounded 
by: (a) the curve, the / (x)-axis, and the X-axis, the limits being 
x = 2 and x = o; (b) the curve, the/ (x)-axis, and the X-axis, the 
limits being x = o and x = I ; (c) the curve and the X-axis, the 
limits being x = 1 and x = 3. 

These examples show the use of the definite integral for finding 
the area of a plane surface. In addition to this application of the 
definite integral, it may be used for finding the areas of curved 
surfaces, the lengths of arcs, the volumes of solids, centers of 
gravity, and moments of inertia. The application of the definite 
integral for finding the length of an arc and for finding the volume 
of a solid of revolution will now be taken up. 



THE LENGTH OF AN ARC 

The Formula. The definite integral can also be used for finding 
the length of an arc. Suppose that it is desired to find the length of 
the arc in the accompanying figure from the point on the curve 
whose abscissa is a to the point whose abscissa is b. Take a small 
segment of the arc at the point whose abscissa is a, and call this 
increment of the arc ds. Then this small arc ds is the hypotenuse of 
the right triangle whose sides are dy and dx. The arc ds is taken so 



78 Elementary Calculus in Senior High-School Mathematics 



very small that there will be no appreciable error if we treat it as 
though it were a straight line. Then ds 2 = dy 2 + dx 2 since the square 
on the hypotenuse is equal to the sum of the squares on the other 
two sides. Under the topic of the differential it was shown that 
dy = f ( X ) dx. Substitute this form of the differential of the func- 
tion in ds 2 = dy 2 + dx 2 . Then the expression becomes 

ds 2 = [f (x) dx] 2 + dx 2 
= [f(x)] 2 .dx 2 + dx 2 
= {[f(*)] 2 -h i}-dx 2 . 

By taking the square roots of 
both sides of this equation, it be- 
comes ds = V f (x) 2 + i- dx. 

This is only an increment of the 
arc. Then the length of the entire 
arc will be the sum of all these in- 
crements as x assumes all the 
values of the range on the X-axis 
between the limits a and b. This 
sum is an integral and is represent- 




ed thus, 



Scale. 1 unit = 2 spaces 

Fig. 24 



i 



V /' (*)■ +i-dx 



Hence to find the length of an 
arc between definite limits, substi- 



,f(X)-axis 



tute the derivative of the function in this formula and then find 
the integral of the resulting expression. 

An Application of the Formula. 
Find the circumference of the 
circle / (x) = •%/ (n — &). 

Divide the circle into four 
quadrants and find the length 
of the arc of one of them. Four 
times this result will give the 
entire circumference. 

The limits are x = o and x = r. 
Hence a = o and b = r. f (x) = 
~ x/f (x) = - x/V (r* -* 2). 
Then/ (x) 2 = x 2 / (r 2 - x 2 ). By 
substituting these values in the 




Fig. 25 

formula, it becomes 



Suggestions for Presentation of Calculus 



79 



- fl 

-L 



I (r 2 - x 2 ) + l] y2 'dx 



x 2 + r s 



']■ 



•dx 



r I (r 2 — x 2 ) y * -dx 

= r [arc sin x lr] r 

= r (arc sin I — arc sin o) 

= r ( 7r/2 — o) = r •* J2. Then the entire cir- 
cumference is four times r- 71-/2, or 2 *■ r. 

Examples. By integration find the circumference of the follow- 
ing circle: x 2 + y 2 = 9. It will be found to be 6 times 3.1416. 

Find the length of an arc of the parabola f (x) = x 2 — 4.x + 5 
with the limits x = 1 and x = 3. Use formula No. 124 in Peirce's 
A Short Table of Integrals, and check by using formulas Nos. 165 
and 160. 

THE VOLUME OF CERTAIN SOLIDS 

The Formula. Suppose that we have given a rectangle OABC 
in the position as shown in the figure. 



t J7xJ-axis 



B 



dx 



Then if we rotate the rectangle 
about 0-X as an axis, the rect- 
angle will generate a cylinder. 

Now use dx as a unit of 
measure and divide OA into as 
many equal parts as dx will be 
contained into it. 

Then pass planes through each 
of these points of division. 

Then these planes will divide 
the cylinder into many equal 
cylinders each with an altitude equal to dx and a base equal to 
the base of the original cylinder. OC, or the radius of this circle, 
or base, is equal to fix). Then the area of this circle is7r[/(x)] 2 , 
since the area of a circle is t times the square of the radius. Hence 
the volume of each cylinder of altitude dx is r.[f (x)] 2 -dx. 

The sum of all such volumes would approximately equal the 
volume of the cylinder, and the limit of this sum as dx approaches 



X 



Fig. 26 



80 Elementary Calculus in Senior High- School Mathematics 

zero as its limit is the volume. This limit is the integral ,/V[/ (x)] 2 dx. 
Hence, the volume of a solid formed by rotating a plane figure 

f b 
about an axis is ir J [f (x)] 2 dx where the upper and lower limits 

of x are b and a respectively. 

Since the generating line CB is parallel to the X-axis, / (x) is 
the same no matter what values are given to x and is always equal 
to OC. Represent OC by a given letter r. Then / (x) = r. 
Represent the length of the cylinder by h, then the limits 
of x are o and h. Substitute these values in the formula 

ir [f (x)] 2 dx. This gives tt r 2 dx. The integral of this is irr 2 x]%. 

The result after substituting the limits is t- r 2 h. This ex- 
pressed in words is the customary rule for finding the volume 

of a cylinder, namely, its volume 
axis is equal to the product of the area 

of the base by the altitude. 

An Application of the Formula. 
Find the volume of a cone of 
radius r and height h. 

The generating line has a con- 
stant slope. It is r /h, and it inter- 
sects the/ (x)-axis at the origin or 
zero. Then in the slope form of 
the straight line, / (x) = mx -\-b, 
m = r /h and b = o. Hence in 

r 
our problem, / (x) = —x. Substitute this value in the formula, 
h 

f b f h r 2 2 

tt [f (x)] 2 dx. This gives ir I -s-xdx. The value of this integral 

Ja Jo ¥ 

is -T-r 2 h. In words, this is the customary rule for finding the volume 

o 
of a cone, namely, the volume of a cone is equal to the product of 
the area of the base times one-third of the altitude. 

Example. Suppose that we have given a semi-circle with radius 
r. This semi-circle rotated about its axis will generate a sphere. 

Find the volume of this sphere. Its volume is ^ -ir.r 3 . 




GENERAL CONCLUSIONS 

From the study as here set forth, it is evident that a first course 
in the elementary calculus is not too difficult for senior high-school 
pupils. This conclusion is based largely upon the following facts: 

1. A historical survey of the natural growth of the calculus in 
the development of mathematics shows that the subject has adapted 
itself slowly to the needs of the race, and that a similar adaptation 
to the needs of those pupils in the senior high schools who care 
to elect the subject is entirely possible; 

2. An examination of those leading European textbooks on 
the elementary calculus which have been written for beginners 
shows that a number of the best teachers of secondary math- 
ematics do not consider the elementary calculus too difficult for 
their pupils; 

3. The theory and the applications of the elementary calculus 
can be so simplified that it becomes an interesting and extremely 
valuable subject for such students in the senior high school as 
show fair ability in mathematics. 

It is therefore evident that a course in the elementary calculus 
should be included in the elective mathematics of the senior high 
school for the following reasons: 

1. It is necessary if our American pupils are to have an equal 
opportunity in mathematics with pupils in schools abroad ; 

2. Such a course is needed to satisfy the demand that has been 
created for additional mathematics in the senior high school by the 
trend of mathematics in our public school system; 

3. The elementary calculus is one of the most important aids 
in applied mathematics and contributes largely to the field of 
pure mathematics ; 

4. Such a course will enable the pupil who intends to enter 
any field of applied mathematics to continue further his study 
of the calculus, as found in such a field, without the aid of a 
teacher. 




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